<!doctype html>
<html>
<head>
<meta charset='UTF-8'><meta name='viewport' content='width=device-width initial-scale=1'>

<link href='https://fonts.loli.net/css?family=Open+Sans:400italic,700italic,700,400&subset=latin,latin-ext' rel='stylesheet' type='text/css' /><style type='text/css'>html {overflow-x: initial !important;}:root { --bg-color:#ffffff; --text-color:#333333; --select-text-bg-color:#B5D6FC; --select-text-font-color:auto; --monospace:"Lucida Console",Consolas,"Courier",monospace; --title-bar-height:20px; }
.mac-os-11 { --title-bar-height:28px; }
html { font-size: 14px; background-color: var(--bg-color); color: var(--text-color); font-family: "Helvetica Neue", Helvetica, Arial, sans-serif; -webkit-font-smoothing: antialiased; }
h1, h2, h3, h4, h5 { white-space: pre-wrap; }
body { margin: 0px; padding: 0px; height: auto; inset: 0px; font-size: 1rem; line-height: 1.42857; overflow-x: hidden; background: inherit; tab-size: 4; }
iframe { margin: auto; }
a.url { word-break: break-all; }
a:active, a:hover { outline: 0px; }
.in-text-selection, ::selection { text-shadow: none; background: var(--select-text-bg-color); color: var(--select-text-font-color); }
#write { margin: 0px auto; height: auto; width: inherit; word-break: normal; overflow-wrap: break-word; position: relative; white-space: normal; overflow-x: visible; padding-top: 36px; }
#write.first-line-indent p { text-indent: 2em; }
#write.first-line-indent li p, #write.first-line-indent p * { text-indent: 0px; }
#write.first-line-indent li { margin-left: 2em; }
.for-image #write { padding-left: 8px; padding-right: 8px; }
body.typora-export { padding-left: 30px; padding-right: 30px; }
.typora-export .footnote-line, .typora-export li, .typora-export p { white-space: pre-wrap; }
.typora-export .task-list-item input { pointer-events: none; }
@media screen and (max-width: 500px) {
  body.typora-export { padding-left: 0px; padding-right: 0px; }
  #write { padding-left: 20px; padding-right: 20px; }
}
#write li > figure:last-child { margin-bottom: 0.5rem; }
#write ol, #write ul { position: relative; }
img { max-width: 100%; vertical-align: middle; image-orientation: from-image; }
button, input, select, textarea { color: inherit; font: inherit; }
input[type="checkbox"], input[type="radio"] { line-height: normal; padding: 0px; }
*, ::after, ::before { box-sizing: border-box; }
#write h1, #write h2, #write h3, #write h4, #write h5, #write h6, #write p, #write pre { width: inherit; }
#write h1, #write h2, #write h3, #write h4, #write h5, #write h6, #write p { position: relative; }
p { line-height: inherit; }
h1, h2, h3, h4, h5, h6 { break-after: avoid-page; break-inside: avoid; orphans: 4; }
p { orphans: 4; }
h1 { font-size: 2rem; }
h2 { font-size: 1.8rem; }
h3 { font-size: 1.6rem; }
h4 { font-size: 1.4rem; }
h5 { font-size: 1.2rem; }
h6 { font-size: 1rem; }
.md-math-block, .md-rawblock, h1, h2, h3, h4, h5, h6, p { margin-top: 1rem; margin-bottom: 1rem; }
.hidden { display: none; }
.md-blockmeta { color: rgb(204, 204, 204); font-weight: 700; font-style: italic; }
a { cursor: pointer; }
sup.md-footnote { padding: 2px 4px; background-color: rgba(238, 238, 238, 0.7); color: rgb(85, 85, 85); border-radius: 4px; cursor: pointer; }
sup.md-footnote a, sup.md-footnote a:hover { color: inherit; text-transform: inherit; text-decoration: inherit; }
#write input[type="checkbox"] { cursor: pointer; width: inherit; height: inherit; }
figure { overflow-x: auto; margin: 1.2em 0px; max-width: calc(100% + 16px); padding: 0px; }
figure > table { margin: 0px; }
thead, tr { break-inside: avoid; break-after: auto; }
thead { display: table-header-group; }
table { border-collapse: collapse; border-spacing: 0px; width: 100%; overflow: auto; break-inside: auto; text-align: left; }
table.md-table td { min-width: 32px; }
.CodeMirror-gutters { border-right: 0px; background-color: inherit; }
.CodeMirror-linenumber { user-select: none; }
.CodeMirror { text-align: left; }
.CodeMirror-placeholder { opacity: 0.3; }
.CodeMirror pre { padding: 0px 4px; }
.CodeMirror-lines { padding: 0px; }
div.hr:focus { cursor: none; }
#write pre { white-space: pre-wrap; }
#write.fences-no-line-wrapping pre { white-space: pre; }
#write pre.ty-contain-cm { white-space: normal; }
.CodeMirror-gutters { margin-right: 4px; }
.md-fences { font-size: 0.9rem; display: block; break-inside: avoid; text-align: left; overflow: visible; white-space: pre; background: inherit; position: relative !important; }
.md-fences-adv-panel { width: 100%; margin-top: 10px; text-align: center; padding-top: 0px; padding-bottom: 8px; overflow-x: auto; }
#write .md-fences.mock-cm { white-space: pre-wrap; }
.md-fences.md-fences-with-lineno { padding-left: 0px; }
#write.fences-no-line-wrapping .md-fences.mock-cm { white-space: pre; overflow-x: auto; }
.md-fences.mock-cm.md-fences-with-lineno { padding-left: 8px; }
.CodeMirror-line, twitterwidget { break-inside: avoid; }
svg { break-inside: avoid; }
.footnotes { opacity: 0.8; font-size: 0.9rem; margin-top: 1em; margin-bottom: 1em; }
.footnotes + .footnotes { margin-top: 0px; }
.md-reset { margin: 0px; padding: 0px; border: 0px; outline: 0px; vertical-align: top; background: 0px 0px; text-decoration: none; text-shadow: none; float: none; position: static; width: auto; height: auto; white-space: nowrap; cursor: inherit; -webkit-tap-highlight-color: transparent; line-height: normal; font-weight: 400; text-align: left; box-sizing: content-box; direction: ltr; }
li div { padding-top: 0px; }
blockquote { margin: 1rem 0px; }
li .mathjax-block, li p { margin: 0.5rem 0px; }
li blockquote { margin: 1rem 0px; }
li { margin: 0px; position: relative; }
blockquote > :last-child { margin-bottom: 0px; }
blockquote > :first-child, li > :first-child { margin-top: 0px; }
.footnotes-area { color: rgb(136, 136, 136); margin-top: 0.714rem; padding-bottom: 0.143rem; white-space: normal; }
#write .footnote-line { white-space: pre-wrap; }
@media print {
  body, html { border: 1px solid transparent; height: 99%; break-after: avoid; break-before: avoid; font-variant-ligatures: no-common-ligatures; }
  #write { margin-top: 0px; border-color: transparent !important; padding-top: 0px !important; padding-bottom: 0px !important; }
  .typora-export * { -webkit-print-color-adjust: exact; }
  .typora-export #write { break-after: avoid; }
  .typora-export #write::after { height: 0px; }
  .is-mac table { break-inside: avoid; }
  #write > p:nth-child(1) { margin-top: 0px; }
  .typora-export-show-outline .typora-export-sidebar { display: none; }
  figure { overflow-x: visible; }
}
.footnote-line { margin-top: 0.714em; font-size: 0.7em; }
a img, img a { cursor: pointer; }
pre.md-meta-block { font-size: 0.8rem; min-height: 0.8rem; white-space: pre-wrap; background: rgb(204, 204, 204); display: block; overflow-x: hidden; }
p > .md-image:only-child:not(.md-img-error) img, p > img:only-child { display: block; margin: auto; }
#write.first-line-indent p > .md-image:only-child:not(.md-img-error) img { left: -2em; position: relative; }
p > .md-image:only-child { display: inline-block; width: 100%; }
#write .MathJax_Display { margin: 0.8em 0px 0px; }
.md-math-block { width: 100%; }
.md-math-block:not(:empty)::after { display: none; }
.MathJax_ref { fill: currentcolor; }
[contenteditable="true"]:active, [contenteditable="true"]:focus, [contenteditable="false"]:active, [contenteditable="false"]:focus { outline: 0px; box-shadow: none; }
.md-task-list-item { position: relative; list-style-type: none; }
.task-list-item.md-task-list-item { padding-left: 0px; }
.md-task-list-item > input { position: absolute; top: 0px; left: 0px; margin-left: -1.2em; margin-top: calc(1em - 10px); border: none; }
.math { font-size: 1rem; }
.md-toc { min-height: 3.58rem; position: relative; font-size: 0.9rem; border-radius: 10px; }
.md-toc-content { position: relative; margin-left: 0px; }
.md-toc-content::after, .md-toc::after { display: none; }
.md-toc-item { display: block; color: rgb(65, 131, 196); }
.md-toc-item a { text-decoration: none; }
.md-toc-inner:hover { text-decoration: underline; }
.md-toc-inner { display: inline-block; cursor: pointer; }
.md-toc-h1 .md-toc-inner { margin-left: 0px; font-weight: 700; }
.md-toc-h2 .md-toc-inner { margin-left: 2em; }
.md-toc-h3 .md-toc-inner { margin-left: 4em; }
.md-toc-h4 .md-toc-inner { margin-left: 6em; }
.md-toc-h5 .md-toc-inner { margin-left: 8em; }
.md-toc-h6 .md-toc-inner { margin-left: 10em; }
@media screen and (max-width: 48em) {
  .md-toc-h3 .md-toc-inner { margin-left: 3.5em; }
  .md-toc-h4 .md-toc-inner { margin-left: 5em; }
  .md-toc-h5 .md-toc-inner { margin-left: 6.5em; }
  .md-toc-h6 .md-toc-inner { margin-left: 8em; }
}
a.md-toc-inner { font-size: inherit; font-style: inherit; font-weight: inherit; line-height: inherit; }
.footnote-line a:not(.reversefootnote) { color: inherit; }
.reversefootnote { font-family: ui-monospace, sans-serif; }
.md-attr { display: none; }
.md-fn-count::after { content: "."; }
code, pre, samp, tt { font-family: var(--monospace); }
kbd { margin: 0px 0.1em; padding: 0.1em 0.6em; font-size: 0.8em; color: rgb(36, 39, 41); background: rgb(255, 255, 255); border: 1px solid rgb(173, 179, 185); border-radius: 3px; box-shadow: rgba(12, 13, 14, 0.2) 0px 1px 0px, rgb(255, 255, 255) 0px 0px 0px 2px inset; white-space: nowrap; vertical-align: middle; }
.md-comment { color: rgb(162, 127, 3); opacity: 0.6; font-family: var(--monospace); }
code { text-align: left; vertical-align: initial; }
a.md-print-anchor { white-space: pre !important; border-width: initial !important; border-style: none !important; border-color: initial !important; display: inline-block !important; position: absolute !important; width: 1px !important; right: 0px !important; outline: 0px !important; background: 0px 0px !important; text-decoration: initial !important; text-shadow: initial !important; }
.os-windows.monocolor-emoji .md-emoji { font-family: "Segoe UI Symbol", sans-serif; }
.md-diagram-panel > svg { max-width: 100%; }
[lang="flow"] svg, [lang="mermaid"] svg { max-width: 100%; height: auto; }
[lang="mermaid"] .node text { font-size: 1rem; }
table tr th { border-bottom: 0px; }
video { max-width: 100%; display: block; margin: 0px auto; }
iframe { max-width: 100%; width: 100%; border: none; }
.highlight td, .highlight tr { border: 0px; }
mark { background: rgb(255, 255, 0); color: rgb(0, 0, 0); }
.md-html-inline .md-plain, .md-html-inline strong, mark .md-inline-math, mark strong { color: inherit; }
.md-expand mark .md-meta { opacity: 0.3 !important; }
mark .md-meta { color: rgb(0, 0, 0); }
@media print {
  .typora-export h1, .typora-export h2, .typora-export h3, .typora-export h4, .typora-export h5, .typora-export h6 { break-inside: avoid; }
}
.md-diagram-panel .messageText { stroke: none !important; }
.md-diagram-panel .start-state { fill: var(--node-fill); }
.md-diagram-panel .edgeLabel rect { opacity: 1 !important; }
.md-fences.md-fences-math { font-size: 1em; }
.md-fences-advanced:not(.md-focus) { padding: 0px; white-space: nowrap; border: 0px; }
.md-fences-advanced:not(.md-focus) { background: inherit; }
.typora-export-show-outline .typora-export-content { max-width: 1440px; margin: auto; display: flex; flex-direction: row; }
.typora-export-sidebar { width: 300px; font-size: 0.8rem; margin-top: 80px; margin-right: 18px; }
.typora-export-show-outline #write { --webkit-flex:2; flex: 2 1 0%; }
.typora-export-sidebar .outline-content { position: fixed; top: 0px; max-height: 100%; overflow: hidden auto; padding-bottom: 30px; padding-top: 60px; width: 300px; }
@media screen and (max-width: 1024px) {
  .typora-export-sidebar, .typora-export-sidebar .outline-content { width: 240px; }
}
@media screen and (max-width: 800px) {
  .typora-export-sidebar { display: none; }
}
.outline-content li, .outline-content ul { margin-left: 0px; margin-right: 0px; padding-left: 0px; padding-right: 0px; list-style: none; overflow-wrap: anywhere; }
.outline-content ul { margin-top: 0px; margin-bottom: 0px; }
.outline-content strong { font-weight: 400; }
.outline-expander { width: 1rem; height: 1.42857rem; position: relative; display: table-cell; vertical-align: middle; cursor: pointer; padding-left: 4px; }
.outline-expander::before { content: ""; position: relative; font-family: Ionicons; display: inline-block; font-size: 8px; vertical-align: middle; }
.outline-item { padding-top: 3px; padding-bottom: 3px; cursor: pointer; }
.outline-expander:hover::before { content: ""; }
.outline-h1 > .outline-item { padding-left: 0px; }
.outline-h2 > .outline-item { padding-left: 1em; }
.outline-h3 > .outline-item { padding-left: 2em; }
.outline-h4 > .outline-item { padding-left: 3em; }
.outline-h5 > .outline-item { padding-left: 4em; }
.outline-h6 > .outline-item { padding-left: 5em; }
.outline-label { cursor: pointer; display: table-cell; vertical-align: middle; text-decoration: none; color: inherit; }
.outline-label:hover { text-decoration: underline; }
.outline-item:hover { border-color: rgb(245, 245, 245); background-color: var(--item-hover-bg-color); }
.outline-item:hover { margin-left: -28px; margin-right: -28px; border-left: 28px solid transparent; border-right: 28px solid transparent; }
.outline-item-single .outline-expander::before, .outline-item-single .outline-expander:hover::before { display: none; }
.outline-item-open > .outline-item > .outline-expander::before { content: ""; }
.outline-children { display: none; }
.info-panel-tab-wrapper { display: none; }
.outline-item-open > .outline-children { display: block; }
.typora-export .outline-item { padding-top: 1px; padding-bottom: 1px; }
.typora-export .outline-item:hover { margin-right: -8px; border-right: 8px solid transparent; }
.typora-export .outline-expander::before { content: "+"; font-family: inherit; top: -1px; }
.typora-export .outline-expander:hover::before, .typora-export .outline-item-open > .outline-item > .outline-expander::before { content: "−"; }
.typora-export-collapse-outline .outline-children { display: none; }
.typora-export-collapse-outline .outline-item-open > .outline-children, .typora-export-no-collapse-outline .outline-children { display: block; }
.typora-export-no-collapse-outline .outline-expander::before { content: "" !important; }
.typora-export-show-outline .outline-item-active > .outline-item .outline-label { font-weight: 700; }
.md-inline-math-container mjx-container { zoom: 0.95; }
mjx-container { break-inside: avoid; }


.CodeMirror { height: auto; }
.CodeMirror.cm-s-inner { background: inherit; }
.CodeMirror-scroll { overflow: auto hidden; z-index: 3; }
.CodeMirror-gutter-filler, .CodeMirror-scrollbar-filler { background-color: rgb(255, 255, 255); }
.CodeMirror-gutters { border-right: 1px solid rgb(221, 221, 221); background: inherit; white-space: nowrap; }
.CodeMirror-linenumber { padding: 0px 3px 0px 5px; text-align: right; color: rgb(153, 153, 153); }
.cm-s-inner .cm-keyword { color: rgb(119, 0, 136); }
.cm-s-inner .cm-atom, .cm-s-inner.cm-atom { color: rgb(34, 17, 153); }
.cm-s-inner .cm-number { color: rgb(17, 102, 68); }
.cm-s-inner .cm-def { color: rgb(0, 0, 255); }
.cm-s-inner .cm-variable { color: rgb(0, 0, 0); }
.cm-s-inner .cm-variable-2 { color: rgb(0, 85, 170); }
.cm-s-inner .cm-variable-3 { color: rgb(0, 136, 85); }
.cm-s-inner .cm-string { color: rgb(170, 17, 17); }
.cm-s-inner .cm-property { color: rgb(0, 0, 0); }
.cm-s-inner .cm-operator { color: rgb(152, 26, 26); }
.cm-s-inner .cm-comment, .cm-s-inner.cm-comment { color: rgb(170, 85, 0); }
.cm-s-inner .cm-string-2 { color: rgb(255, 85, 0); }
.cm-s-inner .cm-meta { color: rgb(85, 85, 85); }
.cm-s-inner .cm-qualifier { color: rgb(85, 85, 85); }
.cm-s-inner .cm-builtin { color: rgb(51, 0, 170); }
.cm-s-inner .cm-bracket { color: rgb(153, 153, 119); }
.cm-s-inner .cm-tag { color: rgb(17, 119, 0); }
.cm-s-inner .cm-attribute { color: rgb(0, 0, 204); }
.cm-s-inner .cm-header, .cm-s-inner.cm-header { color: rgb(0, 0, 255); }
.cm-s-inner .cm-quote, .cm-s-inner.cm-quote { color: rgb(0, 153, 0); }
.cm-s-inner .cm-hr, .cm-s-inner.cm-hr { color: rgb(153, 153, 153); }
.cm-s-inner .cm-link, .cm-s-inner.cm-link { color: rgb(0, 0, 204); }
.cm-negative { color: rgb(221, 68, 68); }
.cm-positive { color: rgb(34, 153, 34); }
.cm-header, .cm-strong { font-weight: 700; }
.cm-del { text-decoration: line-through; }
.cm-em { font-style: italic; }
.cm-link { text-decoration: underline; }
.cm-error { color: red; }
.cm-invalidchar { color: red; }
.cm-constant { color: rgb(38, 139, 210); }
.cm-defined { color: rgb(181, 137, 0); }
div.CodeMirror span.CodeMirror-matchingbracket { color: rgb(0, 255, 0); }
div.CodeMirror span.CodeMirror-nonmatchingbracket { color: rgb(255, 34, 34); }
.cm-s-inner .CodeMirror-activeline-background { background: inherit; }
.CodeMirror { position: relative; overflow: hidden; }
.CodeMirror-scroll { height: 100%; outline: 0px; position: relative; box-sizing: content-box; background: inherit; }
.CodeMirror-sizer { position: relative; }
.CodeMirror-gutter-filler, .CodeMirror-hscrollbar, .CodeMirror-scrollbar-filler, .CodeMirror-vscrollbar { position: absolute; z-index: 6; display: none; outline: 0px; }
.CodeMirror-vscrollbar { right: 0px; top: 0px; overflow: hidden; }
.CodeMirror-hscrollbar { bottom: 0px; left: 0px; overflow: auto hidden; }
.CodeMirror-scrollbar-filler { right: 0px; bottom: 0px; }
.CodeMirror-gutter-filler { left: 0px; bottom: 0px; }
.CodeMirror-gutters { position: absolute; left: 0px; top: 0px; padding-bottom: 10px; z-index: 3; overflow-y: hidden; }
.CodeMirror-gutter { white-space: normal; height: 100%; box-sizing: content-box; padding-bottom: 30px; margin-bottom: -32px; display: inline-block; }
.CodeMirror-gutter-wrapper { position: absolute; z-index: 4; background: 0px 0px !important; border: none !important; }
.CodeMirror-gutter-background { position: absolute; top: 0px; bottom: 0px; z-index: 4; }
.CodeMirror-gutter-elt { position: absolute; cursor: default; z-index: 4; }
.CodeMirror-lines { cursor: text; }
.CodeMirror pre { border-radius: 0px; border-width: 0px; background: 0px 0px; font-family: inherit; font-size: inherit; margin: 0px; white-space: pre; overflow-wrap: normal; color: inherit; z-index: 2; position: relative; overflow: visible; }
.CodeMirror-wrap pre { overflow-wrap: break-word; white-space: pre-wrap; word-break: normal; }
.CodeMirror-code pre { border-right: 30px solid transparent; width: fit-content; }
.CodeMirror-wrap .CodeMirror-code pre { border-right: none; width: auto; }
.CodeMirror-linebackground { position: absolute; inset: 0px; z-index: 0; }
.CodeMirror-linewidget { position: relative; z-index: 2; overflow: auto; }
.CodeMirror-wrap .CodeMirror-scroll { overflow-x: hidden; }
.CodeMirror-measure { position: absolute; width: 100%; height: 0px; overflow: hidden; visibility: hidden; }
.CodeMirror-measure pre { position: static; }
.CodeMirror div.CodeMirror-cursor { position: absolute; visibility: hidden; border-right: none; width: 0px; }
.CodeMirror div.CodeMirror-cursor { visibility: hidden; }
.CodeMirror-focused div.CodeMirror-cursor { visibility: inherit; }
.cm-searching { background: rgba(255, 255, 0, 0.4); }
span.cm-underlined { text-decoration: underline; }
span.cm-strikethrough { text-decoration: line-through; }
.cm-tw-syntaxerror { color: rgb(255, 255, 255); background-color: rgb(153, 0, 0); }
.cm-tw-deleted { text-decoration: line-through; }
.cm-tw-header5 { font-weight: 700; }
.cm-tw-listitem:first-child { padding-left: 10px; }
.cm-tw-box { border-style: solid; border-right-width: 1px; border-bottom-width: 1px; border-left-width: 1px; border-color: inherit; border-top-width: 0px !important; }
.cm-tw-underline { text-decoration: underline; }
@media print {
  .CodeMirror div.CodeMirror-cursor { visibility: hidden; }
}


:root {
    --side-bar-bg-color: #fafafa;
    --control-text-color: #777;
}

@include-when-export url(https://fonts.loli.net/css?family=Open+Sans:400italic,700italic,700,400&subset=latin,latin-ext);

/* open-sans-regular - latin-ext_latin */
  /* open-sans-italic - latin-ext_latin */
    /* open-sans-700 - latin-ext_latin */
    /* open-sans-700italic - latin-ext_latin */
  html {
    font-size: 16px;
    -webkit-font-smoothing: antialiased;
}

body {
    font-family: "Open Sans","Clear Sans", "Helvetica Neue", Helvetica, Arial, 'Segoe UI Emoji', sans-serif;
    color: rgb(51, 51, 51);
    line-height: 1.6;
}

#write {
    max-width: 860px;
  	margin: 0 auto;
  	padding: 30px;
    padding-bottom: 100px;
}

@media only screen and (min-width: 1400px) {
	#write {
		max-width: 1024px;
	}
}

@media only screen and (min-width: 1800px) {
	#write {
		max-width: 1200px;
	}
}

#write > ul:first-child,
#write > ol:first-child{
    margin-top: 30px;
}

a {
    color: #4183C4;
}
h1,
h2,
h3,
h4,
h5,
h6 {
    position: relative;
    margin-top: 1rem;
    margin-bottom: 1rem;
    font-weight: bold;
    line-height: 1.4;
    cursor: text;
}
h1:hover a.anchor,
h2:hover a.anchor,
h3:hover a.anchor,
h4:hover a.anchor,
h5:hover a.anchor,
h6:hover a.anchor {
    text-decoration: none;
}
h1 tt,
h1 code {
    font-size: inherit;
}
h2 tt,
h2 code {
    font-size: inherit;
}
h3 tt,
h3 code {
    font-size: inherit;
}
h4 tt,
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</style><title>Assignment 1</title>
</head>
<body class='typora-export os-windows'><div class='typora-export-content'>
<div id='write'  class=''><h1 id='assignment-1'><span>Assignment 1</span></h1><p><span>Yifan Zhang 8.9.2023</span></p><h2 id='1-image-sampling'><span>1. Image sampling</span></h2><p><span>To solve this problem, we need to calculate the gray level of all pixels in this 5x5 image using the following function:</span></p><div contenteditable="false" spellcheck="false" class="mathjax-block md-end-block md-math-block md-rawblock" id="mathjax-n4" cid="n4" mdtype="math_block" data-math-tag-before="0" data-math-tag-after="0" data-math-labels="[]"><div class="md-rawblock-container md-math-container" tabindex="-1"><mjx-container class="MathJax" jax="SVG" display="true" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="41.39ex" height="2.262ex" role="img" focusable="false" viewBox="0 -750 18294.2 1000" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.566ex;"><defs><path id="MJX-1-TEX-I-1D453" d="M118 -162Q120 -162 124 -164T135 -167T147 -168Q160 -168 171 -155T187 -126Q197 -99 221 27T267 267T289 382V385H242Q195 385 192 387Q188 390 188 397L195 425Q197 430 203 430T250 431Q298 431 298 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xlink:href="#MJX-1-TEX-I-1D466"></use></g><g data-mml-node="mo" transform="translate(16460.4,0)"><use data-c="2264" xlink:href="#MJX-1-TEX-N-2264"></use></g><g data-mml-node="mn" transform="translate(17516.2,0)"><use data-c="31" xlink:href="#MJX-1-TEX-N-31"></use><use data-c="2E" xlink:href="#MJX-1-TEX-N-2E" transform="translate(500,0)"></use></g></g></g></svg><mjx-assistive-mml unselectable="on" display="block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo><mo>=</mo><mi>x</mi><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><mi>y</mi><mo stretchy="false">)</mo><mo>,</mo><mstyle scriptlevel="0"><mspace width="1em"></mspace></mstyle><mn>0</mn><mo>≤</mo><mi>x</mi><mo>≤</mo><mn>1</mn><mo>,</mo><mn>0</mn><mo>≤</mo><mi>y</mi><mo>≤</mo><mn>1.</mn></math></mjx-assistive-mml></mjx-container></div></div><p><span>Since we have divided the image into a 5x5 grid, and x,y∈[0,1], the 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406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z"></path><path id="MJX-53-TEX-N-3D" d="M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z"></path><path id="MJX-53-TEX-N-30" d="M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 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transform="translate(1278,0)"></use></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0.25</mn><mo>×</mo><mn>31</mn><mo>=</mo><mn>7.75</mn></math></mjx-assistive-mml></mjx-container><script type="math/tex">0.25 \times 31 = 7.75</script><span>. However, we need to round it to 8 since the gray level should be an integer. 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xlink:href="#MJX-58-TEX-N-2E" transform="translate(500,0)"></use><use data-c="32" xlink:href="#MJX-58-TEX-N-32" transform="translate(778,0)"></use><use data-c="35" xlink:href="#MJX-58-TEX-N-35" transform="translate(1278,0)"></use></g><g data-mml-node="mo" transform="translate(24881.1,0)"><use data-c="D7" xlink:href="#MJX-58-TEX-N-D7"></use></g><g data-mml-node="mo" transform="translate(25881.3,0)"><use data-c="28" xlink:href="#MJX-58-TEX-N-28"></use></g><g data-mml-node="mn" transform="translate(26270.3,0)"><use data-c="31" xlink:href="#MJX-58-TEX-N-31"></use></g><g data-mml-node="mo" transform="translate(26992.6,0)"><use data-c="2212" xlink:href="#MJX-58-TEX-N-2212"></use></g><g data-mml-node="mn" transform="translate(27992.8,0)"><use data-c="30" xlink:href="#MJX-58-TEX-N-30"></use><use data-c="2E" xlink:href="#MJX-58-TEX-N-2E" transform="translate(500,0)"></use><use data-c="32" xlink:href="#MJX-58-TEX-N-32" transform="translate(778,0)"></use><use data-c="35" xlink:href="#MJX-58-TEX-N-35" transform="translate(1278,0)"></use></g><g data-mml-node="mo" transform="translate(29770.8,0)"><use data-c="29" xlink:href="#MJX-58-TEX-N-29"></use></g><g data-mml-node="mo" transform="translate(30437.6,0)"><use data-c="3D" xlink:href="#MJX-58-TEX-N-3D"></use></g><g data-mml-node="mn" transform="translate(31493.3,0)"><use data-c="30" xlink:href="#MJX-58-TEX-N-30"></use><use data-c="2E" xlink:href="#MJX-58-TEX-N-2E" transform="translate(500,0)"></use><use data-c="31" xlink:href="#MJX-58-TEX-N-31" transform="translate(778,0)"></use><use data-c="38" xlink:href="#MJX-58-TEX-N-38" transform="translate(1278,0)"></use><use data-c="37" xlink:href="#MJX-58-TEX-N-37" transform="translate(1778,0)"></use><use data-c="35" xlink:href="#MJX-58-TEX-N-35" transform="translate(2278,0)"></use></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mi>i</mi><mi>x</mi><mi>e</mi><mi>l</mi><mo stretchy="false">(</mo><mn>0.25</mn><mo>,</mo><mn>0.25</mn><mo stretchy="false">)</mo><mo>:</mo><mi>x</mi><mo>=</mo><mn>0.25</mn><mo>,</mo><mi>y</mi><mo>=</mo><mn>0.25</mn><mo>,</mo><mi>f</mi><mo stretchy="false">(</mo><mn>0.25</mn><mo>,</mo><mn>0.25</mn><mo stretchy="false">)</mo><mo>=</mo><mn>0.25</mn><mo>×</mo><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><mn>0.25</mn><mo stretchy="false">)</mo><mo>=</mo><mn>0.1875</mn></math></mjx-assistive-mml></mjx-container><script type="math/tex">Pixel (0.25, 0.25): x = 0.25, y = 0.25, f(0.25, 0.25) = 0.25\times(1 - 0.25) = 0.1875</script>
<span>Intensity: </span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="20.615ex" height="1.715ex" role="img" focusable="false" viewBox="0 -676 9112 758" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.186ex;"><defs><path id="MJX-59-TEX-N-30" d="M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z"></path><path id="MJX-59-TEX-N-2E" d="M78 60Q78 84 95 102T138 120Q162 120 180 104T199 61Q199 36 182 18T139 0T96 17T78 60Z"></path><path id="MJX-59-TEX-N-31" d="M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z"></path><path id="MJX-59-TEX-N-38" d="M70 417T70 494T124 618T248 666Q319 666 374 624T429 515Q429 485 418 459T392 417T361 389T335 371T324 363L338 354Q352 344 366 334T382 323Q457 264 457 174Q457 95 399 37T249 -22Q159 -22 101 29T43 155Q43 263 172 335L154 348Q133 361 127 368Q70 417 70 494ZM286 386L292 390Q298 394 301 396T311 403T323 413T334 425T345 438T355 454T364 471T369 491T371 513Q371 556 342 586T275 624Q268 625 242 625Q201 625 165 599T128 534Q128 511 141 492T167 463T217 431Q224 426 228 424L286 386ZM250 21Q308 21 350 55T392 137Q392 154 387 169T375 194T353 216T330 234T301 253T274 270Q260 279 244 289T218 306L210 311Q204 311 181 294T133 239T107 157Q107 98 150 60T250 21Z"></path><path id="MJX-59-TEX-N-37" d="M55 458Q56 460 72 567L88 674Q88 676 108 676H128V672Q128 662 143 655T195 646T364 644H485V605L417 512Q408 500 387 472T360 435T339 403T319 367T305 330T292 284T284 230T278 162T275 80Q275 66 275 52T274 28V19Q270 2 255 -10T221 -22Q210 -22 200 -19T179 0T168 40Q168 198 265 368Q285 400 349 489L395 552H302Q128 552 119 546Q113 543 108 522T98 479L95 458V455H55V458Z"></path><path id="MJX-59-TEX-N-35" d="M164 157Q164 133 148 117T109 101H102Q148 22 224 22Q294 22 326 82Q345 115 345 210Q345 313 318 349Q292 382 260 382H254Q176 382 136 314Q132 307 129 306T114 304Q97 304 95 310Q93 314 93 485V614Q93 664 98 664Q100 666 102 666Q103 666 123 658T178 642T253 634Q324 634 389 662Q397 666 402 666Q410 666 410 648V635Q328 538 205 538Q174 538 149 544L139 546V374Q158 388 169 396T205 412T256 420Q337 420 393 355T449 201Q449 109 385 44T229 -22Q148 -22 99 32T50 154Q50 178 61 192T84 210T107 214Q132 214 148 197T164 157Z"></path><path id="MJX-59-TEX-N-D7" d="M630 29Q630 9 609 9Q604 9 587 25T493 118L389 222L284 117Q178 13 175 11Q171 9 168 9Q160 9 154 15T147 29Q147 36 161 51T255 146L359 250L255 354Q174 435 161 449T147 471Q147 480 153 485T168 490Q173 490 175 489Q178 487 284 383L389 278L493 382Q570 459 587 475T609 491Q630 491 630 471Q630 464 620 453T522 355L418 250L522 145Q606 61 618 48T630 29Z"></path><path id="MJX-59-TEX-N-33" d="M127 463Q100 463 85 480T69 524Q69 579 117 622T233 665Q268 665 277 664Q351 652 390 611T430 522Q430 470 396 421T302 350L299 348Q299 347 308 345T337 336T375 315Q457 262 457 175Q457 96 395 37T238 -22Q158 -22 100 21T42 130Q42 158 60 175T105 193Q133 193 151 175T169 130Q169 119 166 110T159 94T148 82T136 74T126 70T118 67L114 66Q165 21 238 21Q293 21 321 74Q338 107 338 175V195Q338 290 274 322Q259 328 213 329L171 330L168 332Q166 335 166 348Q166 366 174 366Q202 366 232 371Q266 376 294 413T322 525V533Q322 590 287 612Q265 626 240 626Q208 626 181 615T143 592T132 580H135Q138 579 143 578T153 573T165 566T175 555T183 540T186 520Q186 498 172 481T127 463Z"></path><path id="MJX-59-TEX-N-3D" d="M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z"></path><path id="MJX-59-TEX-N-32" d="M109 429Q82 429 66 447T50 491Q50 562 103 614T235 666Q326 666 387 610T449 465Q449 422 429 383T381 315T301 241Q265 210 201 149L142 93L218 92Q375 92 385 97Q392 99 409 186V189H449V186Q448 183 436 95T421 3V0H50V19V31Q50 38 56 46T86 81Q115 113 136 137Q145 147 170 174T204 211T233 244T261 278T284 308T305 340T320 369T333 401T340 431T343 464Q343 527 309 573T212 619Q179 619 154 602T119 569T109 550Q109 549 114 549Q132 549 151 535T170 489Q170 464 154 447T109 429Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mn"><use data-c="30" xlink:href="#MJX-59-TEX-N-30"></use><use data-c="2E" xlink:href="#MJX-59-TEX-N-2E" transform="translate(500,0)"></use><use data-c="31" xlink:href="#MJX-59-TEX-N-31" transform="translate(778,0)"></use><use data-c="38" xlink:href="#MJX-59-TEX-N-38" transform="translate(1278,0)"></use><use data-c="37" xlink:href="#MJX-59-TEX-N-37" transform="translate(1778,0)"></use><use data-c="35" xlink:href="#MJX-59-TEX-N-35" transform="translate(2278,0)"></use></g><g data-mml-node="mo" transform="translate(3000.2,0)"><use data-c="D7" xlink:href="#MJX-59-TEX-N-D7"></use></g><g data-mml-node="mn" transform="translate(4000.4,0)"><use data-c="33" xlink:href="#MJX-59-TEX-N-33"></use><use data-c="31" xlink:href="#MJX-59-TEX-N-31" transform="translate(500,0)"></use></g><g data-mml-node="mo" transform="translate(5278.2,0)"><use data-c="3D" xlink:href="#MJX-59-TEX-N-3D"></use></g><g data-mml-node="mn" transform="translate(6334,0)"><use data-c="35" xlink:href="#MJX-59-TEX-N-35"></use><use data-c="2E" xlink:href="#MJX-59-TEX-N-2E" transform="translate(500,0)"></use><use data-c="38" xlink:href="#MJX-59-TEX-N-38" transform="translate(778,0)"></use><use data-c="31" xlink:href="#MJX-59-TEX-N-31" transform="translate(1278,0)"></use><use data-c="32" xlink:href="#MJX-59-TEX-N-32" transform="translate(1778,0)"></use><use data-c="35" xlink:href="#MJX-59-TEX-N-35" transform="translate(2278,0)"></use></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0.1875</mn><mo>×</mo><mn>31</mn><mo>=</mo><mn>5.8125</mn></math></mjx-assistive-mml></mjx-container><script type="math/tex">0.1875 \times 31 = 5.8125 </script><span>(rounded to 6)</span>
<span>Gray level: 6</span></p></li><li><p><span>......</span></p></li><li><p><span>(repeat the process for the remaining pixels)</span></p></li></ul><p><span>By following this process for all pixels, we can obtain the image bellow:</span></p><p><img src="C:\Users\30379\AppData\Roaming\Typora\typora-user-images\image-20230902232217164.png" referrerpolicy="no-referrer" alt="image-20230902232217164"></p><p><span>and the gray level matrix of this image should be:</span></p><p><img src="C:\Users\30379\AppData\Roaming\Typora\typora-user-images\image-20230902232309448.png" referrerpolicy="no-referrer" alt="image-20230902232309448"></p><p><span> </span></p><h2 id='2-histogram-equalization'><span>2. Histogram Equalization</span></h2><p><span>First of all, we derive the transform T(r) and eliminate what </span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="2.047ex" height="1.439ex" role="img" focusable="false" viewBox="0 -442 904.9 636" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.439ex;"><defs><path id="MJX-60-TEX-I-1D45D" d="M23 287Q24 290 25 295T30 317T40 348T55 381T75 411T101 433T134 442Q209 442 230 378L240 387Q302 442 358 442Q423 442 460 395T497 281Q497 173 421 82T249 -10Q227 -10 210 -4Q199 1 187 11T168 28L161 36Q160 35 139 -51T118 -138Q118 -144 126 -145T163 -148H188Q194 -155 194 -157T191 -175Q188 -187 185 -190T172 -194Q170 -194 161 -194T127 -193T65 -192Q-5 -192 -24 -194H-32Q-39 -187 -39 -183Q-37 -156 -26 -148H-6Q28 -147 33 -136Q36 -130 94 103T155 350Q156 355 156 364Q156 405 131 405Q109 405 94 377T71 316T59 280Q57 278 43 278H29Q23 284 23 287ZM178 102Q200 26 252 26Q282 26 310 49T356 107Q374 141 392 215T411 325V331Q411 405 350 405Q339 405 328 402T306 393T286 380T269 365T254 350T243 336T235 326L232 322Q232 321 229 308T218 264T204 212Q178 106 178 102Z"></path><path id="MJX-60-TEX-I-1D45F" d="M21 287Q22 290 23 295T28 317T38 348T53 381T73 411T99 433T132 442Q161 442 183 430T214 408T225 388Q227 382 228 382T236 389Q284 441 347 441H350Q398 441 422 400Q430 381 430 363Q430 333 417 315T391 292T366 288Q346 288 334 299T322 328Q322 376 378 392Q356 405 342 405Q286 405 239 331Q229 315 224 298T190 165Q156 25 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 114 189T154 366Q154 405 128 405Q107 405 92 377T68 316T57 280Q55 278 41 278H27Q21 284 21 287Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="msub"><g data-mml-node="mi"><use data-c="1D45D" xlink:href="#MJX-60-TEX-I-1D45D"></use></g><g data-mml-node="mi" transform="translate(536,-150) scale(0.707)"><use data-c="1D45F" xlink:href="#MJX-60-TEX-I-1D45F"></use></g></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>p</mi><mi>r</mi></msub></math></mjx-assistive-mml></mjx-container><script type="math/tex">p_r</script><span> actually is. Since the number of pixels remains the same before and after the transformation, we can easily obtain:</span></p><div contenteditable="false" spellcheck="false" class="mathjax-block md-end-block md-math-block md-rawblock" id="mathjax-n25" cid="n25" mdtype="math_block" data-math-tag-before="0" data-math-tag-after="0" data-math-labels="[]"><div class="md-rawblock-container md-math-container" tabindex="-1"><mjx-container class="MathJax" jax="SVG" display="true" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="24.492ex" height="5.59ex" role="img" focusable="false" viewBox="0 -1559 10825.3 2470.9" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -2.063ex;"><defs><path id="MJX-2-TEX-LO-222B" d="M114 -798Q132 -824 165 -824H167Q195 -824 223 -764T275 -600T320 -391T362 -164Q365 -143 367 -133Q439 292 523 655T645 1127Q651 1145 655 1157T672 1201T699 1257T733 1306T777 1346T828 1360Q884 1360 912 1325T944 1245Q944 1220 932 1205T909 1186T887 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326Q204 301 223 291T278 274T330 259Q396 230 396 163Q396 135 385 107T352 51T289 7T195 -10Q118 -10 86 19T53 87Q53 126 74 143T118 160Q133 160 146 151T160 120Q160 94 142 76T111 58Q109 57 108 57T107 55Q108 52 115 47T146 34T201 27Q237 27 263 38T301 66T318 97T323 122Q323 150 302 164T254 181T195 196T148 231Q131 256 131 289Z"></path><path id="MJX-2-TEX-N-28" d="M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z"></path><path id="MJX-2-TEX-N-29" d="M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z"></path><path id="MJX-2-TEX-I-1D451" d="M366 683Q367 683 438 688T511 694Q523 694 523 686Q523 679 450 384T375 83T374 68Q374 26 402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487H491Q506 153 506 145Q506 140 503 129Q490 79 473 48T445 8T417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157Q33 205 53 255T101 341Q148 398 195 420T280 442Q336 442 364 400Q369 394 369 396Q370 400 396 505T424 616Q424 629 417 632T378 637H357Q351 643 351 645T353 664Q358 683 366 683ZM352 326Q329 405 277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q233 26 290 98L298 109L352 326Z"></path><path id="MJX-2-TEX-N-3D" d="M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z"></path><path id="MJX-2-TEX-I-1D45F" d="M21 287Q22 290 23 295T28 317T38 348T53 381T73 411T99 433T132 442Q161 442 183 430T214 408T225 388Q227 382 228 382T236 389Q284 441 347 441H350Q398 441 422 400Q430 381 430 363Q430 333 417 315T391 292T366 288Q346 288 334 299T322 328Q322 376 378 392Q356 405 342 405Q286 405 239 331Q229 315 224 298T190 165Q156 25 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 114 189T154 366Q154 405 128 405Q107 405 92 377T68 316T57 280Q55 278 41 278H27Q21 284 21 287Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="msubsup"><g data-mml-node="mo" transform="translate(0 1)"><use data-c="222B" xlink:href="#MJX-2-TEX-LO-222B"></use></g><g data-mml-node="mn" transform="translate(1046.4,1088.1) scale(0.707)"><use data-c="31" xlink:href="#MJX-2-TEX-N-31"></use></g><g data-mml-node="mn" transform="translate(589,-896.4) scale(0.707)"><use data-c="30" xlink:href="#MJX-2-TEX-N-30"></use></g></g><g data-mml-node="msub" transform="translate(1616.6,0)"><g data-mml-node="mi"><use data-c="1D45D" xlink:href="#MJX-2-TEX-I-1D45D"></use></g><g data-mml-node="mi" transform="translate(536,-150) scale(0.707)"><use data-c="1D460" xlink:href="#MJX-2-TEX-I-1D460"></use></g></g><g data-mml-node="mo" transform="translate(2534.3,0)"><use data-c="28" xlink:href="#MJX-2-TEX-N-28"></use></g><g data-mml-node="mi" transform="translate(2923.3,0)"><use data-c="1D460" xlink:href="#MJX-2-TEX-I-1D460"></use></g><g data-mml-node="mo" transform="translate(3392.3,0)"><use data-c="29" xlink:href="#MJX-2-TEX-N-29"></use></g><g data-mml-node="mi" transform="translate(3781.3,0)"><use data-c="1D451" xlink:href="#MJX-2-TEX-I-1D451"></use></g><g data-mml-node="mi" transform="translate(4301.3,0)"><use data-c="1D460" xlink:href="#MJX-2-TEX-I-1D460"></use></g><g data-mml-node="mo" transform="translate(5048,0)"><use data-c="3D" xlink:href="#MJX-2-TEX-N-3D"></use></g><g data-mml-node="msubsup" transform="translate(6103.8,0)"><g data-mml-node="mo" transform="translate(0 1)"><use data-c="222B" xlink:href="#MJX-2-TEX-LO-222B"></use></g><g data-mml-node="mn" transform="translate(1046.4,1088.1) scale(0.707)"><use data-c="31" xlink:href="#MJX-2-TEX-N-31"></use></g><g data-mml-node="mn" transform="translate(589,-896.4) scale(0.707)"><use data-c="30" xlink:href="#MJX-2-TEX-N-30"></use></g></g><g data-mml-node="msub" transform="translate(7720.4,0)"><g data-mml-node="mi"><use data-c="1D45D" xlink:href="#MJX-2-TEX-I-1D45D"></use></g><g data-mml-node="mi" transform="translate(536,-150) scale(0.707)"><use data-c="1D45F" xlink:href="#MJX-2-TEX-I-1D45F"></use></g></g><g data-mml-node="mo" transform="translate(8625.3,0)"><use data-c="28" xlink:href="#MJX-2-TEX-N-28"></use></g><g data-mml-node="mi" transform="translate(9014.3,0)"><use data-c="1D45F" xlink:href="#MJX-2-TEX-I-1D45F"></use></g><g data-mml-node="mo" transform="translate(9465.3,0)"><use data-c="29" xlink:href="#MJX-2-TEX-N-29"></use></g><g data-mml-node="mi" transform="translate(9854.3,0)"><use data-c="1D451" xlink:href="#MJX-2-TEX-I-1D451"></use></g><g data-mml-node="mi" transform="translate(10374.3,0)"><use data-c="1D45F" xlink:href="#MJX-2-TEX-I-1D45F"></use></g></g></g></svg><mjx-assistive-mml unselectable="on" display="block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><msubsup><mo data-mjx-texclass="OP">∫</mo><mn>0</mn><mn>1</mn></msubsup><msub><mi>p</mi><mi>s</mi></msub><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><mi>d</mi><mi>s</mi><mo>=</mo><msubsup><mo data-mjx-texclass="OP">∫</mo><mn>0</mn><mn>1</mn></msubsup><msub><mi>p</mi><mi>r</mi></msub><mo stretchy="false">(</mo><mi>r</mi><mo stretchy="false">)</mo><mi>d</mi><mi>r</mi></math></mjx-assistive-mml></mjx-container></div></div><p><span>Removing the integral sign, the following equation holds.</span></p><div contenteditable="false" spellcheck="false" class="mathjax-block md-end-block md-math-block md-rawblock" id="mathjax-n27" cid="n27" mdtype="math_block" data-math-tag-before="0" data-math-tag-after="0" data-math-labels="[]"><div class="md-rawblock-container md-math-container" tabindex="-1"><mjx-container class="MathJax" jax="SVG" display="true" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="17.177ex" height="2.262ex" role="img" focusable="false" viewBox="0 -750 7592.1 1000" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.566ex;"><defs><path id="MJX-3-TEX-I-1D45D" d="M23 287Q24 290 25 295T30 317T40 348T55 381T75 411T101 433T134 442Q209 442 230 378L240 387Q302 442 358 442Q423 442 460 395T497 281Q497 173 421 82T249 -10Q227 -10 210 -4Q199 1 187 11T168 28L161 36Q160 35 139 -51T118 -138Q118 -144 126 -145T163 -148H188Q194 -155 194 -157T191 -175Q188 -187 185 -190T172 -194Q170 -194 161 -194T127 -193T65 -192Q-5 -192 -24 -194H-32Q-39 -187 -39 -183Q-37 -156 -26 -148H-6Q28 -147 33 -136Q36 -130 94 103T155 350Q156 355 156 364Q156 405 131 405Q109 405 94 377T71 316T59 280Q57 278 43 278H29Q23 284 23 287ZM178 102Q200 26 252 26Q282 26 310 49T356 107Q374 141 392 215T411 325V331Q411 405 350 405Q339 405 328 402T306 393T286 380T269 365T254 350T243 336T235 326L232 322Q232 321 229 308T218 264T204 212Q178 106 178 102Z"></path><path id="MJX-3-TEX-I-1D460" d="M131 289Q131 321 147 354T203 415T300 442Q362 442 390 415T419 355Q419 323 402 308T364 292Q351 292 340 300T328 326Q328 342 337 354T354 372T367 378Q368 378 368 379Q368 382 361 388T336 399T297 405Q249 405 227 379T204 326Q204 301 223 291T278 274T330 259Q396 230 396 163Q396 135 385 107T352 51T289 7T195 -10Q118 -10 86 19T53 87Q53 126 74 143T118 160Q133 160 146 151T160 120Q160 94 142 76T111 58Q109 57 108 57T107 55Q108 52 115 47T146 34T201 27Q237 27 263 38T301 66T318 97T323 122Q323 150 302 164T254 181T195 196T148 231Q131 256 131 289Z"></path><path id="MJX-3-TEX-N-28" d="M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z"></path><path id="MJX-3-TEX-N-29" d="M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z"></path><path id="MJX-3-TEX-I-1D451" d="M366 683Q367 683 438 688T511 694Q523 694 523 686Q523 679 450 384T375 83T374 68Q374 26 402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487H491Q506 153 506 145Q506 140 503 129Q490 79 473 48T445 8T417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157Q33 205 53 255T101 341Q148 398 195 420T280 442Q336 442 364 400Q369 394 369 396Q370 400 396 505T424 616Q424 629 417 632T378 637H357Q351 643 351 645T353 664Q358 683 366 683ZM352 326Q329 405 277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q233 26 290 98L298 109L352 326Z"></path><path id="MJX-3-TEX-N-3D" d="M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z"></path><path id="MJX-3-TEX-I-1D45F" d="M21 287Q22 290 23 295T28 317T38 348T53 381T73 411T99 433T132 442Q161 442 183 430T214 408T225 388Q227 382 228 382T236 389Q284 441 347 441H350Q398 441 422 400Q430 381 430 363Q430 333 417 315T391 292T366 288Q346 288 334 299T322 328Q322 376 378 392Q356 405 342 405Q286 405 239 331Q229 315 224 298T190 165Q156 25 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 114 189T154 366Q154 405 128 405Q107 405 92 377T68 316T57 280Q55 278 41 278H27Q21 284 21 287Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="msub"><g data-mml-node="mi"><use data-c="1D45D" xlink:href="#MJX-3-TEX-I-1D45D"></use></g><g data-mml-node="mi" transform="translate(536,-150) scale(0.707)"><use data-c="1D460" xlink:href="#MJX-3-TEX-I-1D460"></use></g></g><g data-mml-node="mo" transform="translate(917.6,0)"><use data-c="28" xlink:href="#MJX-3-TEX-N-28"></use></g><g data-mml-node="mi" transform="translate(1306.6,0)"><use data-c="1D460" xlink:href="#MJX-3-TEX-I-1D460"></use></g><g data-mml-node="mo" transform="translate(1775.6,0)"><use data-c="29" xlink:href="#MJX-3-TEX-N-29"></use></g><g data-mml-node="mi" transform="translate(2164.6,0)"><use data-c="1D451" xlink:href="#MJX-3-TEX-I-1D451"></use></g><g data-mml-node="mi" transform="translate(2684.6,0)"><use data-c="1D460" xlink:href="#MJX-3-TEX-I-1D460"></use></g><g data-mml-node="mo" transform="translate(3431.4,0)"><use data-c="3D" xlink:href="#MJX-3-TEX-N-3D"></use></g><g data-mml-node="msub" transform="translate(4487.2,0)"><g data-mml-node="mi"><use data-c="1D45D" xlink:href="#MJX-3-TEX-I-1D45D"></use></g><g data-mml-node="mi" transform="translate(536,-150) scale(0.707)"><use data-c="1D45F" xlink:href="#MJX-3-TEX-I-1D45F"></use></g></g><g data-mml-node="mo" transform="translate(5392.1,0)"><use data-c="28" xlink:href="#MJX-3-TEX-N-28"></use></g><g data-mml-node="mi" transform="translate(5781.1,0)"><use data-c="1D45F" xlink:href="#MJX-3-TEX-I-1D45F"></use></g><g data-mml-node="mo" transform="translate(6232.1,0)"><use data-c="29" xlink:href="#MJX-3-TEX-N-29"></use></g><g data-mml-node="mi" transform="translate(6621.1,0)"><use data-c="1D451" xlink:href="#MJX-3-TEX-I-1D451"></use></g><g data-mml-node="mi" transform="translate(7141.1,0)"><use data-c="1D45F" xlink:href="#MJX-3-TEX-I-1D45F"></use></g></g></g></svg><mjx-assistive-mml unselectable="on" display="block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><msub><mi>p</mi><mi>s</mi></msub><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><mi>d</mi><mi>s</mi><mo>=</mo><msub><mi>p</mi><mi>r</mi></msub><mo stretchy="false">(</mo><mi>r</mi><mo stretchy="false">)</mo><mi>d</mi><mi>r</mi></math></mjx-assistive-mml></mjx-container></div></div><p><span>Substituting </span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="6.224ex" height="1.946ex" role="img" focusable="false" viewBox="0 -666 2751.2 860" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.439ex;"><defs><path id="MJX-61-TEX-I-1D45D" d="M23 287Q24 290 25 295T30 317T40 348T55 381T75 411T101 433T134 442Q209 442 230 378L240 387Q302 442 358 442Q423 442 460 395T497 281Q497 173 421 82T249 -10Q227 -10 210 -4Q199 1 187 11T168 28L161 36Q160 35 139 -51T118 -138Q118 -144 126 -145T163 -148H188Q194 -155 194 -157T191 -175Q188 -187 185 -190T172 -194Q170 -194 161 -194T127 -193T65 -192Q-5 -192 -24 -194H-32Q-39 -187 -39 -183Q-37 -156 -26 -148H-6Q28 -147 33 -136Q36 -130 94 103T155 350Q156 355 156 364Q156 405 131 405Q109 405 94 377T71 316T59 280Q57 278 43 278H29Q23 284 23 287ZM178 102Q200 26 252 26Q282 26 310 49T356 107Q374 141 392 215T411 325V331Q411 405 350 405Q339 405 328 402T306 393T286 380T269 365T254 350T243 336T235 326L232 322Q232 321 229 308T218 264T204 212Q178 106 178 102Z"></path><path id="MJX-61-TEX-I-1D460" d="M131 289Q131 321 147 354T203 415T300 442Q362 442 390 415T419 355Q419 323 402 308T364 292Q351 292 340 300T328 326Q328 342 337 354T354 372T367 378Q368 378 368 379Q368 382 361 388T336 399T297 405Q249 405 227 379T204 326Q204 301 223 291T278 274T330 259Q396 230 396 163Q396 135 385 107T352 51T289 7T195 -10Q118 -10 86 19T53 87Q53 126 74 143T118 160Q133 160 146 151T160 120Q160 94 142 76T111 58Q109 57 108 57T107 55Q108 52 115 47T146 34T201 27Q237 27 263 38T301 66T318 97T323 122Q323 150 302 164T254 181T195 196T148 231Q131 256 131 289Z"></path><path id="MJX-61-TEX-N-3D" d="M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z"></path><path id="MJX-61-TEX-N-31" d="M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="msub"><g data-mml-node="mi"><use data-c="1D45D" xlink:href="#MJX-61-TEX-I-1D45D"></use></g><g data-mml-node="mi" transform="translate(536,-150) scale(0.707)"><use data-c="1D460" xlink:href="#MJX-61-TEX-I-1D460"></use></g></g><g data-mml-node="mo" transform="translate(1195.4,0)"><use data-c="3D" xlink:href="#MJX-61-TEX-N-3D"></use></g><g data-mml-node="mn" transform="translate(2251.2,0)"><use data-c="31" xlink:href="#MJX-61-TEX-N-31"></use></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>p</mi><mi>s</mi></msub><mo>=</mo><mn>1</mn></math></mjx-assistive-mml></mjx-container><script type="math/tex">p_s=1</script><span> and 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transform="translate(746.8,0)"><use data-c="3D" xlink:href="#MJX-62-TEX-N-3D"></use></g><g data-mml-node="mi" transform="translate(1802.6,0)"><use data-c="1D447" xlink:href="#MJX-62-TEX-I-1D447"></use></g><g data-mml-node="mo" transform="translate(2506.6,0)"><use data-c="28" xlink:href="#MJX-62-TEX-N-28"></use></g><g data-mml-node="mi" transform="translate(2895.6,0)"><use data-c="1D45F" xlink:href="#MJX-62-TEX-I-1D45F"></use></g><g data-mml-node="mo" transform="translate(3346.6,0)"><use data-c="29" xlink:href="#MJX-62-TEX-N-29"></use></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>s</mi><mo>=</mo><mi>T</mi><mo stretchy="false">(</mo><mi>r</mi><mo stretchy="false">)</mo></math></mjx-assistive-mml></mjx-container><script type="math/tex"> s= T(r)</script><span> into the above equation.</span></p><div contenteditable="false" spellcheck="false" class="mathjax-block md-end-block md-math-block md-rawblock" id="mathjax-n29" cid="n29" mdtype="math_block" data-math-tag-before="0" data-math-tag-after="0" data-math-labels="[]"><div class="md-rawblock-container md-math-container" tabindex="-1"><mjx-container class="MathJax" jax="SVG" display="true" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="20.846ex" height="2.262ex" role="img" focusable="false" viewBox="0 -750 9214 1000" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.566ex;"><defs><path id="MJX-4-TEX-I-1D451" d="M366 683Q367 683 438 688T511 694Q523 694 523 686Q523 679 450 384T375 83T374 68Q374 26 402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487H491Q506 153 506 145Q506 140 503 129Q490 79 473 48T445 8T417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157Q33 205 53 255T101 341Q148 398 195 420T280 442Q336 442 364 400Q369 394 369 396Q370 400 396 505T424 616Q424 629 417 632T378 637H357Q351 643 351 645T353 664Q358 683 366 683ZM352 326Q329 405 277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q233 26 290 98L298 109L352 326Z"></path><path id="MJX-4-TEX-I-1D460" d="M131 289Q131 321 147 354T203 415T300 442Q362 442 390 415T419 355Q419 323 402 308T364 292Q351 292 340 300T328 326Q328 342 337 354T354 372T367 378Q368 378 368 379Q368 382 361 388T336 399T297 405Q249 405 227 379T204 326Q204 301 223 291T278 274T330 259Q396 230 396 163Q396 135 385 107T352 51T289 7T195 -10Q118 -10 86 19T53 87Q53 126 74 143T118 160Q133 160 146 151T160 120Q160 94 142 76T111 58Q109 57 108 57T107 55Q108 52 115 47T146 34T201 27Q237 27 263 38T301 66T318 97T323 122Q323 150 302 164T254 181T195 196T148 231Q131 256 131 289Z"></path><path id="MJX-4-TEX-N-3D" d="M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z"></path><path id="MJX-4-TEX-I-1D447" d="M40 437Q21 437 21 445Q21 450 37 501T71 602L88 651Q93 669 101 677H569H659Q691 677 697 676T704 667Q704 661 687 553T668 444Q668 437 649 437Q640 437 637 437T631 442L629 445Q629 451 635 490T641 551Q641 586 628 604T573 629Q568 630 515 631Q469 631 457 630T439 622Q438 621 368 343T298 60Q298 48 386 46Q418 46 427 45T436 36Q436 31 433 22Q429 4 424 1L422 0Q419 0 415 0Q410 0 363 1T228 2Q99 2 64 0H49Q43 6 43 9T45 27Q49 40 55 46H83H94Q174 46 189 55Q190 56 191 56Q196 59 201 76T241 233Q258 301 269 344Q339 619 339 625Q339 630 310 630H279Q212 630 191 624Q146 614 121 583T67 467Q60 445 57 441T43 437H40Z"></path><path id="MJX-4-TEX-N-28" d="M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z"></path><path id="MJX-4-TEX-I-1D45F" d="M21 287Q22 290 23 295T28 317T38 348T53 381T73 411T99 433T132 442Q161 442 183 430T214 408T225 388Q227 382 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-193T65 -192Q-5 -192 -24 -194H-32Q-39 -187 -39 -183Q-37 -156 -26 -148H-6Q28 -147 33 -136Q36 -130 94 103T155 350Q156 355 156 364Q156 405 131 405Q109 405 94 377T71 316T59 280Q57 278 43 278H29Q23 284 23 287ZM178 102Q200 26 252 26Q282 26 310 49T356 107Q374 141 392 215T411 325V331Q411 405 350 405Q339 405 328 402T306 393T286 380T269 365T254 350T243 336T235 326L232 322Q232 321 229 308T218 264T204 212Q178 106 178 102Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><use data-c="1D451" xlink:href="#MJX-4-TEX-I-1D451"></use></g><g data-mml-node="mi" transform="translate(520,0)"><use data-c="1D460" xlink:href="#MJX-4-TEX-I-1D460"></use></g><g data-mml-node="mo" transform="translate(1266.8,0)"><use data-c="3D" xlink:href="#MJX-4-TEX-N-3D"></use></g><g data-mml-node="mi" transform="translate(2322.6,0)"><use data-c="1D451" xlink:href="#MJX-4-TEX-I-1D451"></use></g><g data-mml-node="mi" transform="translate(2842.6,0)"><use data-c="1D447" xlink:href="#MJX-4-TEX-I-1D447"></use></g><g data-mml-node="mo" transform="translate(3546.6,0)"><use data-c="28" xlink:href="#MJX-4-TEX-N-28"></use></g><g data-mml-node="mi" transform="translate(3935.6,0)"><use data-c="1D45F" xlink:href="#MJX-4-TEX-I-1D45F"></use></g><g data-mml-node="mo" transform="translate(4386.6,0)"><use data-c="29" xlink:href="#MJX-4-TEX-N-29"></use></g><g data-mml-node="mo" transform="translate(5053.3,0)"><use data-c="3D" xlink:href="#MJX-4-TEX-N-3D"></use></g><g data-mml-node="msub" transform="translate(6109.1,0)"><g data-mml-node="mi"><use data-c="1D45D" xlink:href="#MJX-4-TEX-I-1D45D"></use></g><g data-mml-node="mi" transform="translate(536,-150) scale(0.707)"><use data-c="1D45F" xlink:href="#MJX-4-TEX-I-1D45F"></use></g></g><g data-mml-node="mo" transform="translate(7014,0)"><use data-c="28" xlink:href="#MJX-4-TEX-N-28"></use></g><g data-mml-node="mi" transform="translate(7403,0)"><use data-c="1D45F" xlink:href="#MJX-4-TEX-I-1D45F"></use></g><g data-mml-node="mo" transform="translate(7854,0)"><use data-c="29" xlink:href="#MJX-4-TEX-N-29"></use></g><g data-mml-node="mi" transform="translate(8243,0)"><use data-c="1D451" xlink:href="#MJX-4-TEX-I-1D451"></use></g><g data-mml-node="mi" transform="translate(8763,0)"><use data-c="1D45F" xlink:href="#MJX-4-TEX-I-1D45F"></use></g></g></g></svg><mjx-assistive-mml unselectable="on" display="block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>d</mi><mi>s</mi><mo>=</mo><mi>d</mi><mi>T</mi><mo stretchy="false">(</mo><mi>r</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mi>p</mi><mi>r</mi></msub><mo stretchy="false">(</mo><mi>r</mi><mo stretchy="false">)</mo><mi>d</mi><mi>r</mi></math></mjx-assistive-mml></mjx-container></div></div><p><span>s can be easily obtained by substituting the function </span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="23.212ex" height="2.262ex" role="img" focusable="false" viewBox="0 -750 10259.8 1000" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.566ex;"><defs><path id="MJX-63-TEX-I-1D45D" d="M23 287Q24 290 25 295T30 317T40 348T55 381T75 411T101 433T134 442Q209 442 230 378L240 387Q302 442 358 442Q423 442 460 395T497 281Q497 173 421 82T249 -10Q227 -10 210 -4Q199 1 187 11T168 28L161 36Q160 35 139 -51T118 -138Q118 -144 126 -145T163 -148H188Q194 -155 194 -157T191 -175Q188 -187 185 -190T172 -194Q170 -194 161 -194T127 -193T65 -192Q-5 -192 -24 -194H-32Q-39 -187 -39 -183Q-37 -156 -26 -148H-6Q28 -147 33 -136Q36 -130 94 103T155 350Q156 355 156 364Q156 405 131 405Q109 405 94 377T71 316T59 280Q57 278 43 278H29Q23 284 23 287ZM178 102Q200 26 252 26Q282 26 310 49T356 107Q374 141 392 215T411 325V331Q411 405 350 405Q339 405 328 402T306 393T286 380T269 365T254 350T243 336T235 326L232 322Q232 321 229 308T218 264T204 212Q178 106 178 102Z"></path><path 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stretchy="false">]</mo></math></mjx-assistive-mml></mjx-container><script type="math/tex">p_r = 6r(1-r), r∈[0,1]</script></p><div contenteditable="false" spellcheck="false" class="mathjax-block md-end-block md-math-block md-rawblock" id="mathjax-n31" cid="n31" mdtype="math_block" data-math-tag-before="0" data-math-tag-after="0" data-math-labels="[]"><div class="md-rawblock-container md-math-container" tabindex="-1"><mjx-container class="MathJax" jax="SVG" display="true" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="34.532ex" height="5.232ex" role="img" focusable="false" viewBox="0 -1400.6 15263.1 2312.5" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -2.063ex;"><defs><path id="MJX-5-TEX-I-1D446" d="M308 24Q367 24 416 76T466 197Q466 260 414 284Q308 311 278 321T236 341Q176 383 176 462Q176 523 208 573T273 648Q302 673 343 688T407 704H418H425Q521 704 564 640Q565 640 577 653T603 682T623 704Q624 704 627 704T632 705Q645 705 645 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61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><use data-c="1D453" xlink:href="#MJX-67-TEX-I-1D453"></use></g><g data-mml-node="mo" transform="translate(550,0)"><use data-c="28" xlink:href="#MJX-67-TEX-N-28"></use></g><g data-mml-node="mi" transform="translate(939,0)"><use data-c="1D456" xlink:href="#MJX-67-TEX-I-1D456"></use></g><g data-mml-node="mo" transform="translate(1284,0)"><use data-c="2C" xlink:href="#MJX-67-TEX-N-2C"></use></g><g data-mml-node="mi" transform="translate(1728.7,0)"><use data-c="1D457" xlink:href="#MJX-67-TEX-I-1D457"></use></g><g data-mml-node="mo" transform="translate(2140.7,0)"><use data-c="29" xlink:href="#MJX-67-TEX-N-29"></use></g><g data-mml-node="mo" transform="translate(2807.4,0)"><use data-c="3E" xlink:href="#MJX-67-TEX-N-3E"></use></g><g data-mml-node="mn" transform="translate(3863.2,0)"><use data-c="31" xlink:href="#MJX-67-TEX-N-31"></use></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo stretchy="false">(</mo><mi>i</mi><mo>,</mo><mi>j</mi><mo stretchy="false">)</mo><mo>&gt;</mo><mn>1</mn></math></mjx-assistive-mml></mjx-container><script type="math/tex">f (i, j) > 1</script><span>.</span></p></blockquote><p><img src = "E:\隆德大学LUND\2023 period 1\Image Analysis\Assignment 1\figure\微信图片_20230901175331.jpg" style="zoom:43%;"></p><p><span>I used MATLAB to threshold the image, so g should be:</span></p><p><img src="C:\Users\30379\AppData\Roaming\Typora\typora-user-images\image-20230901175541209.png" style="zoom:60%;" ></p><blockquote><p><span>b) Find the 8-connected components for g = 1 (as defined in the lectures).</span></p></blockquote><p><span>8-connected components in matrix form:</span></p><p><img src="C:\Users\30379\AppData\Roaming\Typora\typora-user-images\image-20230901175930753.png" style="zoom:60%;" ></p><p><span>Using MATLAB to fill in color and display in image form:</span></p><p><img src="E:\隆德大学LUND\2023 period 1\Image Analysis\Assignment 1\figure\figure 3.jpg"></p><p><span>Here is my MATLAB code, only the custom_bwlabel function I wrote, due to the limitation of space:</span></p><pre class="md-fences md-end-block md-fences-with-lineno ty-contain-cm modeLoaded" spellcheck="false" lang="matlab" style="break-inside: unset;"><div class="CodeMirror cm-s-inner cm-s-null-scroll CodeMirror-wrap" lang="matlab"><div style="overflow: hidden; position: relative; width: 3px; height: 0px; top: 9.51875px; left: 53px;"><textarea autocorrect="off" autocapitalize="off" spellcheck="false" tabindex="0" style="position: absolute; bottom: -1em; padding: 0px; width: 1000px; height: 1em; outline: none;"></textarea></div><div class="CodeMirror-scrollbar-filler" cm-not-content="true"></div><div class="CodeMirror-gutter-filler" cm-not-content="true"></div><div class="CodeMirror-scroll" tabindex="-1"><div class="CodeMirror-sizer" style="margin-left: 45px; margin-bottom: 0px; border-right-width: 0px; padding-right: 0px; padding-bottom: 0px;"><div style="position: relative; top: 0px;"><div class="CodeMirror-lines" role="presentation"><div role="presentation" style="position: relative; outline: none;"><div class="CodeMirror-measure"><div class="CodeMirror-linenumber CodeMirror-gutter-elt"><div>111</div></div></div><div class="CodeMirror-measure"></div><div style="position: relative; z-index: 1;"></div><div class="CodeMirror-code" role="presentation" style=""><div style="position: relative;" class="CodeMirror-activeline"><div class="CodeMirror-activeline-background CodeMirror-linebackground"></div><div class="CodeMirror-gutter-background CodeMirror-activeline-gutter" style="left: -45px; width: 45px;"></div><div class="CodeMirror-gutter-wrapper CodeMirror-activeline-gutter" style="left: -45px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt CodeMirror-linenumber-show" style="left: 0px; width: 36px;">1</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-builtin">function</span> [<span class="cm-variable">pixelCoordinates</span>, <span class="cm-variable">labels</span>, <span class="cm-variable">numLabels</span>] = <span class="cm-variable">custom_bwlabel</span>(<span class="cm-variable">image</span>, <span class="cm-variable">c</span>)</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -45px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 36px;">2</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp;  [<span class="cm-variable">rows</span>, <span class="cm-variable">cols</span>] = <span class="cm-builtin">size</span>(<span class="cm-variable">image</span>);</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -45px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 36px;">3</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp;<span class="cm-variable">labels</span> = <span class="cm-builtin">zeros</span>(<span class="cm-variable">rows</span>, <span class="cm-variable">cols</span>); &nbsp;<span class="cm-comment">% Create a zero matrix of the same size as the original image</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -45px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 36px;">4</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp;<span class="cm-variable">nextLabel</span> = <span class="cm-number">1</span>; &nbsp;<span class="cm-comment">% Initialize the next available label</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -45px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 36px;">5</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span cm-text="" cm-zwsp="">
</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -45px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 36px;">6</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp;<span class="cm-comment">% Traverse through the image pixels</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -45px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 36px;">7</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp;<span class="cm-keyword">for</span> <span class="cm-variable">i</span> = <span class="cm-number">1</span>:<span class="cm-variable">rows</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -45px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 36px;">8</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp; &nbsp; &nbsp;<span class="cm-keyword">for</span> <span class="cm-variable">j</span> = <span class="cm-number">1</span>:<span class="cm-variable">cols</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -45px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 36px;">9</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;<span class="cm-comment">% If the current pixel is foreground and unmarked</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -45px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt CodeMirror-linenumber-show" style="left: 0px; width: 36px;">10</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;<span class="cm-keyword">if</span> <span class="cm-variable">image</span>(<span class="cm-variable">i</span>, <span class="cm-variable">j</span>) <span class="cm-operator">~</span>= <span class="cm-number">0</span> <span class="cm-operator">&amp;&amp;</span> <span class="cm-variable">labels</span>(<span class="cm-variable">i</span>, <span class="cm-variable">j</span>) <span class="cm-operator">==</span> <span class="cm-number">0</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -45px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 36px;">11</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;<span class="cm-comment">% Perform depth-first search for connected component labeling</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -45px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 36px;">12</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;<span class="cm-variable">labels</span> = <span class="cm-variable">dfs</span>(<span class="cm-variable">image</span>, <span class="cm-variable">labels</span>, <span class="cm-variable">i</span>, <span class="cm-variable">j</span>, <span class="cm-variable">nextLabel</span>, <span class="cm-variable">c</span>);</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -45px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 36px;">13</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;<span class="cm-variable">nextLabel</span> = <span class="cm-variable">nextLabel</span> <span class="cm-operator">+</span> <span class="cm-number">1</span>; &nbsp;<span class="cm-comment">% Update the next available label</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -45px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 36px;">14</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;<span class="cm-keyword">end</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -45px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 36px;">15</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp; &nbsp; &nbsp;<span class="cm-keyword">end</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -45px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 36px;">16</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp;<span class="cm-keyword">end</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -45px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 36px;">17</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span cm-text="" cm-zwsp="">
</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -45px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 36px;">18</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp;<span class="cm-variable">numLabels</span> = <span class="cm-variable">nextLabel</span> <span class="cm-operator">-</span> <span class="cm-number">1</span>; &nbsp;<span class="cm-comment">% Calculate the number of connected components</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -45px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 36px;">19</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp;<span class="cm-variable">pixelCoordinates</span> = <span class="cm-variable">cell</span>(<span class="cm-number">1</span>, <span class="cm-variable">numLabels</span>);</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -45px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt CodeMirror-linenumber-show" style="left: 0px; width: 36px;">20</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span cm-text="" cm-zwsp="">
</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -45px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 36px;">21</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp;<span class="cm-comment">% Traverse through all pixels</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -45px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 36px;">22</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp;<span class="cm-keyword">for</span> <span class="cm-variable">row</span> = <span class="cm-number">1</span>:<span class="cm-variable">rows</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -45px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 36px;">23</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp; &nbsp; &nbsp;<span class="cm-keyword">for</span> <span class="cm-variable">col</span> = <span class="cm-number">1</span>:<span class="cm-variable">cols</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -45px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 36px;">24</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;<span class="cm-comment">% Get the label of the current pixel</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -45px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 36px;">25</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;<span class="cm-variable">label</span> = <span class="cm-variable">labels</span>(<span class="cm-variable">row</span>, <span class="cm-variable">col</span>);</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -45px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 36px;">26</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -45px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 36px;">27</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;<span class="cm-comment">% If the current pixel belongs to a connected component</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -45px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 36px;">28</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;<span class="cm-keyword">if</span> <span class="cm-variable">label</span> <span class="cm-operator">&gt;</span> <span class="cm-number">0</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -45px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 36px;">29</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;<span class="cm-comment">% Add the coordinates of the current pixel to the corresponding connected component's vector</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -45px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt CodeMirror-linenumber-show" style="left: 0px; width: 36px;">30</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;<span class="cm-variable">pixelCoordinates</span>{<span class="cm-variable">label</span>} = [<span class="cm-variable">pixelCoordinates</span>{<span class="cm-variable">label</span>}; [<span class="cm-variable">row</span>, <span class="cm-variable">col</span>]];</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -45px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 36px;">31</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;<span class="cm-keyword">end</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -45px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 36px;">32</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp; &nbsp; &nbsp;<span class="cm-keyword">end</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -45px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 36px;">33</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp;<span class="cm-keyword">end</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -45px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 36px;">34</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span cm-text="" cm-zwsp="">
</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -45px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 36px;">35</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp;<span class="cm-comment">% Create an empty cell array to store the top-left pixel coordinates of each connected component</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -45px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 36px;">36</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp;<span class="cm-variable">regionPositions</span> = <span class="cm-variable">cell</span>(<span class="cm-number">1</span>, <span class="cm-variable">numLabels</span>);</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -45px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 36px;">37</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span cm-text="" cm-zwsp="">
</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -45px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 36px;">38</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp;<span class="cm-comment">% Traverse through all pixels and record the top-left pixel coordinates of each connected component</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -45px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 36px;">39</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp;<span class="cm-keyword">for</span> <span class="cm-variable">row</span> = <span class="cm-number">1</span>:<span class="cm-variable">rows</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -45px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt CodeMirror-linenumber-show" style="left: 0px; width: 36px;">40</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp; &nbsp; &nbsp;<span class="cm-keyword">for</span> <span class="cm-variable">col</span> = <span class="cm-number">1</span>:<span class="cm-variable">cols</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -45px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 36px;">41</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;<span class="cm-variable">label</span> = <span class="cm-variable">labels</span>(<span class="cm-variable">row</span>, <span class="cm-variable">col</span>);</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -45px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 36px;">42</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;<span class="cm-keyword">if</span> <span class="cm-variable">label</span> <span class="cm-operator">&gt;</span> <span class="cm-number">0</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -45px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 36px;">43</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;<span class="cm-comment">% If the top-left pixel coordinate of the current pixel's connected component is not recorded yet, record it</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -45px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 36px;">44</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;<span class="cm-keyword">if</span> <span class="cm-variable">isempty</span>(<span class="cm-variable">regionPositions</span>{<span class="cm-variable">label</span>})</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -45px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 36px;">45</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;<span class="cm-variable">regionPositions</span>{<span class="cm-variable">label</span>} = [<span class="cm-variable">row</span>, <span class="cm-variable">col</span>];</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -45px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 36px;">46</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;<span class="cm-keyword">end</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -45px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 36px;">47</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;<span class="cm-keyword">end</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -45px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 36px;">48</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp; &nbsp; &nbsp;<span class="cm-keyword">end</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -45px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 36px;">49</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp;<span class="cm-keyword">end</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -45px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt CodeMirror-linenumber-show" style="left: 0px; width: 36px;">50</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span cm-text="" cm-zwsp="">
</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -45px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 36px;">51</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp;<span class="cm-comment">% Sort the connected components based on the x-coordinate of the top-left pixel</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -45px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 36px;">52</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp;  [<span class="cm-operator">~</span>, <span class="cm-variable">sortedIndices</span>] = <span class="cm-variable">sort</span>(<span class="cm-builtin">cellfun</span>(<span class="cm-operator">@</span>(<span class="cm-variable">x</span>) <span class="cm-variable">x</span>(<span class="cm-number">2</span>), <span class="cm-variable">regionPositions</span>));</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -45px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 36px;">53</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp;<span class="cm-comment">% Create a new cell array to store the sorted coordinates of the connected components</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -45px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 36px;">54</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp;<span class="cm-variable">sortedPixelCoordinates</span> = <span class="cm-variable">cell</span>(<span class="cm-number">1</span>, <span class="cm-variable">numLabels</span>);</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -45px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 36px;">55</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span cm-text="" cm-zwsp="">
</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -45px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 36px;">56</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp;<span class="cm-comment">% Adjust the order of the connected component coordinates according to the sorted indices</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -45px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 36px;">57</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp;<span class="cm-keyword">for</span> <span class="cm-variable">i</span> = <span class="cm-number">1</span>:<span class="cm-variable">numLabels</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -45px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 36px;">58</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp; &nbsp; &nbsp;<span class="cm-variable">label</span> = <span class="cm-variable">sortedIndices</span>(<span class="cm-variable">i</span>);</span></pre></div><div style="position: relative;" class=""><div class="CodeMirror-gutter-wrapper" style="left: -45px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 36px;">59</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp; &nbsp; &nbsp;<span class="cm-variable">sortedPixelCoordinates</span>{<span class="cm-variable">i</span>} = <span class="cm-variable">pixelCoordinates</span>{<span class="cm-variable">label</span>};</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -45px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt CodeMirror-linenumber-show" style="left: 0px; width: 36px;">60</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp;<span class="cm-keyword">end</span></span></pre></div><div class="" style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -45px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 36px;">61</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span cm-text="" cm-zwsp="">
</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -45px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 36px;">62</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp;<span class="cm-comment">% Replace the original pixelCoordinates with the sorted coordinates</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -45px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 36px;">63</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp;<span class="cm-variable">pixelCoordinates</span> = <span class="cm-variable">sortedPixelCoordinates</span>;</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -45px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 36px;">64</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-keyword">end</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -45px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 36px;">65</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span cm-text="" cm-zwsp="">
</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -45px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 36px;">66</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span cm-text="" cm-zwsp="">
</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -45px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 36px;">67</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-builtin">function</span> <span class="cm-variable">labels</span> = <span class="cm-variable">dfs</span>(<span class="cm-variable">image</span>, <span class="cm-variable">labels</span>, <span class="cm-variable">row</span>, <span class="cm-variable">col</span>, <span class="cm-variable">label</span>, <span class="cm-variable">c</span>)</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -45px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 36px;">68</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp;  [<span class="cm-variable">rows</span>, <span class="cm-variable">cols</span>] = <span class="cm-builtin">size</span>(<span class="cm-variable">image</span>);</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -45px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 36px;">69</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp;<span class="cm-variable">stack</span> = <span class="cm-builtin">zeros</span>(<span class="cm-variable">rows</span> <span class="cm-operator">*</span> <span class="cm-variable">cols</span>, <span class="cm-number">2</span>); &nbsp;<span class="cm-comment">% Create a stack for implementing depth-first search</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -45px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt CodeMirror-linenumber-show" style="left: 0px; width: 36px;">70</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp;<span class="cm-variable">stackSize</span> = <span class="cm-number">1</span>;</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -45px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 36px;">71</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp;<span class="cm-variable">stack</span>(<span class="cm-number">1</span>, :) = [<span class="cm-variable">row</span>, <span class="cm-variable">col</span>];</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -45px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 36px;">72</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp;<span class="cm-variable">labels</span>(<span class="cm-variable">row</span>, <span class="cm-variable">col</span>) = <span class="cm-variable">label</span>;</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -45px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 36px;">73</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span cm-text="" cm-zwsp="">
</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -45px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 36px;">74</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp;<span class="cm-keyword">while</span> <span class="cm-variable">stackSize</span> <span class="cm-operator">&gt;</span> <span class="cm-number">0</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -45px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 36px;">75</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp; &nbsp; &nbsp;<span class="cm-variable">current</span> = <span class="cm-variable">stack</span>(<span class="cm-variable">stackSize</span>, :);</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -45px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 36px;">76</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp; &nbsp; &nbsp;<span class="cm-variable">stackSize</span> = <span class="cm-variable">stackSize</span> <span class="cm-operator">-</span> <span class="cm-number">1</span>;</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -45px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 36px;">77</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span cm-text="" cm-zwsp="">
</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -45px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 36px;">78</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp; &nbsp; &nbsp;<span class="cm-comment">% Check the four neighbors (4-connectivity) of the current pixel</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -45px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 36px;">79</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp; &nbsp; &nbsp;<span class="cm-variable">neighbors_4</span> = [<span class="cm-variable">current</span>(<span class="cm-number">1</span>)<span class="cm-number">-1</span>, <span class="cm-variable">current</span>(<span class="cm-number">2</span>);</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -45px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt CodeMirror-linenumber-show" style="left: 0px; width: 36px;">80</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; <span class="cm-variable">current</span>(<span class="cm-number">1</span>)<span class="cm-number">+1</span>, <span class="cm-variable">current</span>(<span class="cm-number">2</span>);</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -45px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 36px;">81</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; <span class="cm-variable">current</span>(<span class="cm-number">1</span>), <span class="cm-variable">current</span>(<span class="cm-number">2</span>)<span class="cm-number">-1</span>;</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -45px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 36px;">82</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; <span class="cm-variable">current</span>(<span class="cm-number">1</span>), <span class="cm-variable">current</span>(<span class="cm-number">2</span>)<span class="cm-number">+1</span>];</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -45px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 36px;">83</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp; &nbsp; &nbsp;<span class="cm-comment">% 8-connectivity</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -45px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 36px;">84</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp; &nbsp; &nbsp;<span class="cm-variable">neighbors_8</span> = [<span class="cm-variable">current</span>(<span class="cm-number">1</span>)<span class="cm-number">-1</span>, <span class="cm-variable">current</span>(<span class="cm-number">2</span>);</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -45px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 36px;">85</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; <span class="cm-variable">current</span>(<span class="cm-number">1</span>)<span class="cm-number">+1</span>, <span class="cm-variable">current</span>(<span class="cm-number">2</span>);</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -45px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 36px;">86</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; <span class="cm-variable">current</span>(<span class="cm-number">1</span>), <span class="cm-variable">current</span>(<span class="cm-number">2</span>)<span class="cm-number">-1</span>;</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -45px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 36px;">87</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; <span class="cm-variable">current</span>(<span class="cm-number">1</span>), <span class="cm-variable">current</span>(<span class="cm-number">2</span>)<span class="cm-number">+1</span>;</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -45px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 36px;">88</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; <span class="cm-variable">current</span>(<span class="cm-number">1</span>)<span class="cm-number">-1</span>, <span class="cm-variable">current</span>(<span class="cm-number">2</span>)<span class="cm-number">-1</span>;</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -45px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 36px;">89</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; <span class="cm-variable">current</span>(<span class="cm-number">1</span>)<span class="cm-number">-1</span>, <span class="cm-variable">current</span>(<span class="cm-number">2</span>)<span class="cm-number">+1</span>;</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -45px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt CodeMirror-linenumber-show" style="left: 0px; width: 36px;">90</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; <span class="cm-variable">current</span>(<span class="cm-number">1</span>)<span class="cm-number">+1</span>, <span class="cm-variable">current</span>(<span class="cm-number">2</span>)<span class="cm-number">-1</span>;</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -45px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 36px;">91</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; <span class="cm-variable">current</span>(<span class="cm-number">1</span>)<span class="cm-number">+1</span>, <span class="cm-variable">current</span>(<span class="cm-number">2</span>)<span class="cm-number">+1</span>]; &nbsp; &nbsp;</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -45px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 36px;">92</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp; &nbsp; &nbsp;</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -45px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 36px;">93</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp; &nbsp; &nbsp;<span class="cm-keyword">if</span> <span class="cm-variable">c</span> <span class="cm-operator">==</span> <span class="cm-number">4</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -45px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 36px;">94</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;<span class="cm-variable">neighbors</span> = <span class="cm-variable">neighbors_4</span>;</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -45px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 36px;">95</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp; &nbsp; &nbsp;<span class="cm-keyword">elseif</span> <span class="cm-variable">c</span> <span class="cm-operator">==</span> <span class="cm-number">8</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -45px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 36px;">96</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;<span class="cm-variable">neighbors</span> = <span class="cm-variable">neighbors_8</span>;</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -45px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 36px;">97</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp; &nbsp; &nbsp;<span class="cm-keyword">end</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -45px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 36px;">98</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span cm-text="" cm-zwsp="">
</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -45px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 36px;">99</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp; &nbsp; &nbsp;<span class="cm-keyword">for</span> <span class="cm-variable">k</span> = <span class="cm-number">1</span>:<span class="cm-builtin">size</span>(<span class="cm-variable">neighbors</span>, <span class="cm-number">1</span>)</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -45px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt CodeMirror-linenumber-show" style="left: 0px; width: 36px;">100</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;<span class="cm-variable">neighborRow</span> = <span class="cm-variable">neighbors</span>(<span class="cm-variable">k</span>, <span class="cm-number">1</span>);</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -45px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 36px;">101</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;<span class="cm-variable">neighborCol</span> = <span class="cm-variable">neighbors</span>(<span class="cm-variable">k</span>, <span class="cm-number">2</span>);</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -45px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 36px;">102</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span cm-text="" cm-zwsp="">
</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -45px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 36px;">103</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;<span class="cm-comment">% Check if the neighbor is within the image range, its pixel value is non-zero, and it is unmarked</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -45px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 36px;">104</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;<span class="cm-keyword">if</span> <span class="cm-variable">neighborRow</span> <span class="cm-operator">&gt;</span>= <span class="cm-number">1</span> <span class="cm-operator">&amp;&amp;</span> <span class="cm-variable">neighborRow</span> <span class="cm-operator">&lt;</span>= <span class="cm-variable">rows</span> <span class="cm-operator">&amp;&amp;</span> <span class="cm-variable">neighborCol</span> <span class="cm-operator">&gt;</span>= <span class="cm-number">1</span> <span class="cm-operator">&amp;&amp;</span> <span class="cm-variable">neighborCol</span> <span class="cm-operator">&lt;</span>= <span class="cm-variable">cols</span> <span class="cm-operator">&amp;&amp;</span> <span class="cm-variable">image</span>(<span class="cm-variable">neighborRow</span>, <span class="cm-variable">neighborCol</span>) <span class="cm-operator">~</span>= <span class="cm-number">0</span> <span class="cm-operator">&amp;&amp;</span> <span class="cm-variable">labels</span>(<span class="cm-variable">neighborRow</span>, <span class="cm-variable">neighborCol</span>) <span class="cm-operator">==</span> <span class="cm-number">0</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -45px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 36px;">105</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;<span class="cm-variable">stackSize</span> = <span class="cm-variable">stackSize</span> <span class="cm-operator">+</span> <span class="cm-number">1</span>;</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -45px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 36px;">106</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;<span class="cm-variable">stack</span>(<span class="cm-variable">stackSize</span>, :)= [<span class="cm-variable">neighborRow</span>, <span class="cm-variable">neighborCol</span>];</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -45px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 36px;">107</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;<span class="cm-variable">labels</span>(<span class="cm-variable">neighborRow</span>, <span class="cm-variable">neighborCol</span>) = <span class="cm-variable">label</span>;</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -45px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 36px;">108</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;<span class="cm-keyword">end</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -45px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 36px;">109</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp; &nbsp; &nbsp;<span class="cm-keyword">end</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -45px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt CodeMirror-linenumber-show" style="left: 0px; width: 36px;">110</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp;<span class="cm-keyword">end</span></span></pre></div><div style="position: relative;" class=""><div class="CodeMirror-gutter-wrapper" style="left: -45px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt CodeMirror-linenumber-show" style="left: 0px; width: 36px;">111</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-keyword">end</span></span></pre></div></div></div></div></div></div><div style="position: absolute; height: 0px; width: 1px; border-bottom: 0px solid transparent; top: 2580px;"></div><div class="CodeMirror-gutters" style="height: 2580px; left: 0px;"><div class="CodeMirror-gutter CodeMirror-linenumbers" style="width: 44px;"></div></div></div></div></pre><p>&nbsp;</p><h2 id='4-segmentation-part-of-ocr'><span>4. Segmentation part of OCR</span></h2><p><img src = "C:\Users\30379\AppData\Roaming\Typora\typora-user-images\image-20230903120410389.png" style="zoom:60%;"></p><p><img src = "C:\Users\30379\AppData\Roaming\Typora\typora-user-images\image-20230903120318583.png" style="zoom:60%;"></p><p><img src = "C:\Users\30379\AppData\Roaming\Typora\typora-user-images\image-20230903120651232.png" style="zoom:60%;"></p><p><img src = "C:\Users\30379\AppData\Roaming\Typora\typora-user-images\image-20230903120727873.png" style="zoom:60%;"></p><p><img src = "C:\Users\30379\AppData\Roaming\Typora\typora-user-images\image-20230903120808199.png" style="zoom:60%;"></p><p><img src = "C:\Users\30379\AppData\Roaming\Typora\typora-user-images\image-20230903120837220.png" style="zoom:60%;"></p><p><img src = "C:\Users\30379\AppData\Roaming\Typora\typora-user-images\image-20230903121020457.png" style="zoom:60%;"></p><p><img src = "C:\Users\30379\AppData\Roaming\Typora\typora-user-images\image-20230907143036862.png" style="zoom:60%;"></p><p><span>MATLAB code</span></p><pre class="md-fences md-end-block md-fences-with-lineno ty-contain-cm modeLoaded" spellcheck="false" lang="matlab" style="break-inside: unset;"><div class="CodeMirror cm-s-inner cm-s-null-scroll CodeMirror-wrap" lang="matlab"><div style="overflow: hidden; position: relative; width: 3px; height: 0px; top: 9.51875px; left: 44px;"><textarea autocorrect="off" autocapitalize="off" spellcheck="false" tabindex="0" style="position: absolute; bottom: -1em; padding: 0px; width: 1000px; height: 1em; outline: none;"></textarea></div><div class="CodeMirror-scrollbar-filler" cm-not-content="true"></div><div class="CodeMirror-gutter-filler" cm-not-content="true"></div><div class="CodeMirror-scroll" tabindex="-1"><div class="CodeMirror-sizer" style="margin-left: 36px; margin-bottom: 0px; border-right-width: 0px; padding-right: 0px; padding-bottom: 0px;"><div style="position: relative; top: 0px;"><div class="CodeMirror-lines" role="presentation"><div role="presentation" style="position: relative; outline: none;"><div class="CodeMirror-measure"><div class="CodeMirror-linenumber CodeMirror-gutter-elt"><div>66</div></div></div><div class="CodeMirror-measure"></div><div style="position: relative; z-index: 1;"></div><div class="CodeMirror-code" role="presentation" style=""><div class="CodeMirror-activeline" style="position: relative;"><div class="CodeMirror-activeline-background CodeMirror-linebackground"></div><div class="CodeMirror-gutter-background CodeMirror-activeline-gutter" style="left: -36px; width: 36px;"></div><div class="CodeMirror-gutter-wrapper CodeMirror-activeline-gutter" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt CodeMirror-linenumber-show" style="left: 0px; width: 27px;">1</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-builtin">function</span> <span class="cm-variable">S</span> = <span class="cm-variable">im2segment</span>(<span class="cm-variable">img</span>)</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">2</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-variable">DISPLAY</span> = <span class="cm-number">0</span>;</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">3</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-variable">img</span> = <span class="cm-variable">uint8</span>(<span class="cm-variable">img</span>);</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">4</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span cm-text="" cm-zwsp="">
</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">5</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-variable">img</span> = <span class="cm-variable">imbilatfilt</span>(<span class="cm-variable">img</span>);</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">6</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span cm-text="" cm-zwsp="">
</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">7</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-variable">threshold</span> = <span class="cm-number">0.135</span>; <span class="cm-comment">% Set the threshold of image binarize, default 0.135</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">8</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-variable">binary_image</span> = <span class="cm-variable">imbinarize</span>(<span class="cm-variable">img</span>, <span class="cm-variable">threshold</span>); <span class="cm-comment">% binarize</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">9</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-keyword">if</span> <span class="cm-variable">DISPLAY</span> <span class="cm-operator">==</span> <span class="cm-number">1</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt CodeMirror-linenumber-show" style="left: 0px; width: 27px;">10</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-tab" role="presentation" cm-text="	">    </span><span class="cm-variable">figure</span>(<span class="cm-number">1</span>);</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">11</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-tab" role="presentation" cm-text="	">    </span><span class="cm-variable">imshow</span>(<span class="cm-variable">binary_image</span>);</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">12</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-keyword">end</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">13</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span cm-text="" cm-zwsp="">
</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">14</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-comment">% Specify the standard deviation of the Gaussian filter</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">15</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-variable">sigma</span> = <span class="cm-number">0.46</span>; <span class="cm-comment">% The larger the value of sigma, the stronger the smoothing effect</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">16</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span cm-text="" cm-zwsp="">
</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">17</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-comment">% Apply Gaussian filtering to the image</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">18</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-variable">img</span> = <span class="cm-variable">imgaussfilt</span>(<span class="cm-variable">img</span>, <span class="cm-variable">sigma</span>);</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">19</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span cm-text="" cm-zwsp="">
</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt CodeMirror-linenumber-show" style="left: 0px; width: 27px;">20</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-comment">%% I rewrite the bwlabel, so I am actually using 24-connect component.</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">21</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;">[<span class="cm-variable">pixelCoordinates</span>, <span class="cm-variable">labeled_Image</span>, <span class="cm-variable">numLabels</span>] = <span class="cm-variable">custom_bwlabel</span>(<span class="cm-variable">binary_image</span>, <span class="cm-number">24</span>);</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">22</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span cm-text="" cm-zwsp="">
</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">23</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-comment">%% filtering the connected components, determine the number of elements in a cell</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">24</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-keyword">if</span> <span class="cm-variable">numLabels</span> <span class="cm-operator">&gt;</span> <span class="cm-number">5</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">25</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp;<span class="cm-comment">% get the dimensions of the element</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">26</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp;<span class="cm-variable">dimensions</span> = <span class="cm-builtin">cellfun</span>(<span class="cm-operator">@</span><span class="cm-variable">numel</span>, <span class="cm-variable">pixelCoordinates</span>);</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">27</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp;</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">28</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp;<span class="cm-comment">% Remove the vector with the smallest dimension until only five elements are left.</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">29</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp;<span class="cm-variable">numToRemove</span> = <span class="cm-variable">numLabels</span> <span class="cm-operator">-</span> <span class="cm-number">5</span>;</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt CodeMirror-linenumber-show" style="left: 0px; width: 27px;">30</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp;  [<span class="cm-operator">~</span>, <span class="cm-variable">sortedIndices</span>] = <span class="cm-variable">sort</span>(<span class="cm-variable">dimensions</span>);</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">31</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp;<span class="cm-variable">removeIndices</span> = <span class="cm-variable">sortedIndices</span>(<span class="cm-number">1</span>:<span class="cm-variable">numToRemove</span>);</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">32</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp;<span class="cm-variable">pixelCoordinates</span>(<span class="cm-variable">removeIndices</span>) = [];</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">33</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-keyword">end</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">34</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span cm-text="" cm-zwsp="">
</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">35</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-comment">%% display the labeled image</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">36</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-keyword">if</span> <span class="cm-variable">DISPALY</span> <span class="cm-operator">==</span> <span class="cm-number">1</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">37</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-tab" role="presentation" cm-text="	">    </span><span class="cm-variable">figure</span>(<span class="cm-number">2</span>);</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">38</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-tab" role="presentation" cm-text="	">    </span><span class="cm-variable">imshow</span>(<span class="cm-variable">label2rgb</span>(<span class="cm-variable">labeled_Image</span>, <span class="cm-string">'hsv'</span>, <span class="cm-string">'k'</span>, <span class="cm-string">'shuffle'</span>));</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">39</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-tab" role="presentation" cm-text="	">    </span><span class="cm-builtin">title</span>(<span class="cm-string">'Sigmentation'</span>);</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt CodeMirror-linenumber-show" style="left: 0px; width: 27px;">40</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-keyword">end</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">41</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span cm-text="" cm-zwsp="">
</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">42</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-comment">%% Iterate through each connected component</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">43</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-keyword">for</span> <span class="cm-variable">label</span> = <span class="cm-number">1</span>:<span class="cm-number">5</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">44</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-tab" role="presentation" cm-text="	">    </span><span class="cm-comment">% create an all-black image with the same size as the input image.</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">45</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp;<span class="cm-variable">regionImage</span> = <span class="cm-builtin">zeros</span>(<span class="cm-builtin">size</span>(<span class="cm-variable">img</span>));</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">46</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp;<span class="cm-comment">% Retrieve the pixel indices of the current connected component</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">47</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp;<span class="cm-variable">coordinates</span> = <span class="cm-variable">pixelCoordinates</span>{<span class="cm-variable">label</span>};</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">48</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp;</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">49</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp; <span class="cm-comment">% Set the pixels of the current connected component to white</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt CodeMirror-linenumber-show" style="left: 0px; width: 27px;">50</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp;<span class="cm-keyword">for</span> <span class="cm-variable">i</span> = <span class="cm-number">1</span>:<span class="cm-builtin">size</span>(<span class="cm-variable">coordinates</span>, <span class="cm-number">1</span>)</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">51</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp; &nbsp; &nbsp;<span class="cm-variable">row</span> = <span class="cm-variable">coordinates</span>(<span class="cm-variable">i</span>, <span class="cm-number">1</span>);</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">52</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp; &nbsp; &nbsp;<span class="cm-variable">col</span> = <span class="cm-variable">coordinates</span>(<span class="cm-variable">i</span>, <span class="cm-number">2</span>);</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">53</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp; &nbsp; &nbsp;<span class="cm-variable">regionImage</span>(<span class="cm-variable">row</span>, <span class="cm-variable">col</span>) = <span class="cm-number">1</span>;</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">54</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp;<span class="cm-keyword">end</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">55</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span cm-text="" cm-zwsp="">
</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">56</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp;<span class="cm-variable">S</span>{<span class="cm-variable">label</span>} = <span class="cm-variable">regionImage</span>;</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">57</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp;</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">58</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp;<span class="cm-comment">% display</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">59</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp;<span class="cm-keyword">if</span> <span class="cm-variable">DISPLAY</span> <span class="cm-operator">==</span> <span class="cm-number">1</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt CodeMirror-linenumber-show" style="left: 0px; width: 27px;">60</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp; &nbsp; &nbsp;<span class="cm-variable">figure</span>;</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">61</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp; &nbsp; &nbsp;<span class="cm-variable">imshow</span>(<span class="cm-variable">output_image</span>);</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">62</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp; &nbsp; &nbsp;<span class="cm-builtin">title</span>(<span class="cm-string">'Individual Sigmentation'</span>);</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">63</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp;<span class="cm-keyword">end</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">64</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-keyword">end</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">65</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span cm-text="" cm-zwsp="">
</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt CodeMirror-linenumber-show" style="left: 0px; width: 27px;">66</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-keyword">end</span></span></pre></div></div></div></div></div></div><div style="position: absolute; height: 0px; width: 1px; border-bottom: 0px solid transparent; top: 1520px;"></div><div class="CodeMirror-gutters" style="height: 1520px; left: 0px;"><div class="CodeMirror-gutter CodeMirror-linenumbers" style="width: 35px;"></div></div></div></div></pre><p>&nbsp;</p><h2 id='5-dimensionality'><span>5. Dimensionality</span></h2><blockquote><p><span>A: The set of gray-scale (monochrome) images with 3 × 2 pixels is a vector space. It has a finite dimension k, and we can define a basis that consists of k 3 × 2 images.</span></p></blockquote><ul><li><p><span>A) The set of gray-scale (monochrome) images with 3 × 2 pixels is a vector space. To determine its dimension, we need to count the number of independent basis elements. Since each pixel can take on a grayscale value independently, the total number of pixels in a 3 × 2 image is 6. Therefore, the dimension of this vector space is k = 6.</span></p></li></ul><p><span>       To provide an example basis for A, we need to find k 3 × 2 images that are linearly independent. 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displaystyle="true"><mtr><mtd><msub><mi>e</mi><mn>1</mn></msub><mo>=</mo><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">[</mo><mtable columnspacing="1em" rowspacing="4pt"><mtr><mtd><mn>1</mn></mtd><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd></mtr></mtable><mo data-mjx-texclass="CLOSE">]</mo></mrow><mo>,</mo><msub><mi>e</mi><mn>2</mn></msub><mo>=</mo><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">[</mo><mtable columnspacing="1em" rowspacing="4pt"><mtr><mtd><mn>0</mn></mtd><mtd><mn>1</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd></mtr></mtable><mo data-mjx-texclass="CLOSE">]</mo></mrow><mo>,</mo><msub><mi>e</mi><mn>3</mn></msub><mo>=</mo><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">[</mo><mtable columnspacing="1em" rowspacing="4pt"><mtr><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mn>1</mn></mtd><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd></mtr></mtable><mo data-mjx-texclass="CLOSE">]</mo></mrow><mo>,</mo><msub><mi>e</mi><mn>4</mn></msub><mo>=</mo><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">[</mo><mtable columnspacing="1em" rowspacing="4pt"><mtr><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><mn>1</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd></mtr></mtable><mo data-mjx-texclass="CLOSE">]</mo></mrow><mo>,</mo><msub><mi>e</mi><mn>5</mn></msub><mo>=</mo><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">[</mo><mtable columnspacing="1em" rowspacing="4pt"><mtr><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mn>1</mn></mtd><mtd><mn>0</mn></mtd></mtr></mtable><mo data-mjx-texclass="CLOSE">]</mo></mrow><mo>,</mo><msub><mi>e</mi><mn>6</mn></msub><mo>=</mo><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">[</mo><mtable columnspacing="1em" rowspacing="4pt"><mtr><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><mn>1</mn></mtd></mtr></mtable><mo data-mjx-texclass="CLOSE">]</mo></mrow></mtd></mtr></mtable></math></mjx-assistive-mml></mjx-container></div></div><blockquote><p><span>B: The set of gray-scale (monochrome) images with 1500 × 2000 pixels is a vector space.</span></p></blockquote><ul><li><p><span>B) The set of gray-scale (monochrome) images with 1500 × 2000 pixels is also a vector space. To determine its dimension, we count the number of independent basis elements. In this case, each pixel can take on a grayscale value independently, and the total number of pixels in a 1500 × 2000 image is 3,000,000. Thus, the dimension of this vector space is k = 3,000,000.</span></p></li></ul><p><span>       For B, describing how the basis elements can be chosen is more challenging due to the large number of pixels involved. However, one possible approach is to consider the basis elements as images where each pixel has a value of 1 at a specific location and 0 elsewhere. 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data-mjx-texclass="ORD"><mo>⋯</mo></mrow></mtd><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mo>⋮</mo></mrow></mrow></mtd><mtd><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mo>⋮</mo></mrow></mrow></mtd><mtd><mrow data-mjx-texclass="ORD"><mo>⋱</mo></mrow></mtd><mtd><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mo>⋮</mo></mrow></mrow></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd><mtd><mrow data-mjx-texclass="ORD"><mo>⋯</mo></mrow></mtd><mtd><mn>0</mn></mtd></mtr></mtable><mo data-mjx-texclass="CLOSE">]</mo></mrow><mo>,</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo>,</mo><mi>e</mi><mi>t</mi><mi>c</mi><mo>.</mo></mtd></mtr></mtable></math></mjx-assistive-mml></mjx-container></div></div><p>&nbsp;</p><h2 id='6-scalar-products-and-norm-on-images'><span>6. Scalar products and norm on images</span></h2><p><span>The scalar product, also known as the dot product, is a way to measure the similarity or correlation between two vectors. 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transform="translate(9311,0)"><use data-c="28" xlink:href="#MJX-8-TEX-N-28"></use></g><g data-mml-node="mi" transform="translate(9700,0)"><use data-c="1D456" xlink:href="#MJX-8-TEX-I-1D456"></use></g><g data-mml-node="mo" transform="translate(10045,0)"><use data-c="2C" xlink:href="#MJX-8-TEX-N-2C"></use></g><g data-mml-node="mi" transform="translate(10489.7,0)"><use data-c="1D457" xlink:href="#MJX-8-TEX-I-1D457"></use></g><g data-mml-node="mo" transform="translate(10901.7,0)"><use data-c="29" xlink:href="#MJX-8-TEX-N-29"></use></g><g data-mml-node="mo" transform="translate(11290.7,0)"><use data-c="2E" xlink:href="#MJX-8-TEX-N-2E"></use></g></g></g></svg><mjx-assistive-mml unselectable="on" display="block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>f</mi><mo>⋅</mo><mi>g</mi><mo>=</mo><munderover><mo data-mjx-texclass="OP">∑</mo><mrow data-mjx-texclass="ORD"><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>M</mi></munderover><munderover><mo data-mjx-texclass="OP">∑</mo><mrow data-mjx-texclass="ORD"><mi>j</mi><mo>=</mo><mn>1</mn></mrow><mi>N</mi></munderover><mrow data-mjx-texclass="ORD"><mover><mi>f</mi><mo stretchy="false">¯</mo></mover></mrow><mo stretchy="false">(</mo><mi>i</mi><mo>,</mo><mi>j</mi><mo stretchy="false">)</mo><mi>g</mi><mo stretchy="false">(</mo><mi>i</mi><mo>,</mo><mi>j</mi><mo stretchy="false">)</mo><mo>.</mo></math></mjx-assistive-mml></mjx-container></div></div><p><span>The norm of an image, in this case, refers to the norm of the matrix representing the image.</span></p><div contenteditable="false" spellcheck="false" class="mathjax-block md-end-block md-math-block md-rawblock" id="mathjax-n82" cid="n82" mdtype="math_block" data-math-tag-before="0" data-math-tag-after="0" data-math-labels="[]"><div class="md-rawblock-container md-math-container" tabindex="-1"><mjx-container class="MathJax" jax="SVG" display="true" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="37.978ex" 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867 340T797 59Q797 55 798 54T805 50T822 48T855 46H886Q892 37 892 35Q892 19 885 5Q880 0 869 0Q864 0 828 1T736 2Q675 2 644 2T609 1Q592 1 592 11Q592 13 594 25Q598 41 602 43T625 46Q652 46 685 49Q699 52 704 61Q706 65 742 207T813 490T848 631L654 322Q458 10 453 5Q451 4 449 3Q444 0 433 0Q418 0 415 7Q413 11 374 317L335 624L267 354Q200 88 200 79Q206 46 272 46H282Q288 41 289 37T286 19Q282 3 278 1Q274 0 267 0Q265 0 255 0T221 1T157 2Q127 2 95 1T58 0Q43 0 39 2T35 11Q35 13 38 25T43 40Q45 46 65 46Q135 46 154 86Q158 92 223 354T289 629Z"></path><path id="MJX-9-TEX-I-1D457" d="M297 596Q297 627 318 644T361 661Q378 661 389 651T403 623Q403 595 384 576T340 557Q322 557 310 567T297 596ZM288 376Q288 405 262 405Q240 405 220 393T185 362T161 325T144 293L137 279Q135 278 121 278H107Q101 284 101 286T105 299Q126 348 164 391T252 441Q253 441 260 441T272 442Q296 441 316 432Q341 418 354 401T367 348V332L318 133Q267 -67 264 -75Q246 -125 194 -164T75 -204Q25 -204 7 -183T-12 -137Q-12 -110 7 -91T53 -71Q70 -71 82 -81T95 -112Q95 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data-mml-node="mo" transform="translate(278,0) translate(0 -0.5)"><use data-c="7C" xlink:href="#MJX-9-TEX-N-7C"></use></g><g data-mml-node="mi" transform="translate(556,0)"><use data-c="1D453" xlink:href="#MJX-9-TEX-I-1D453"></use></g><g data-mml-node="mo" transform="translate(1106,0) translate(0 -0.5)"><use data-c="7C" xlink:href="#MJX-9-TEX-N-7C"></use></g><g data-mml-node="mo" transform="translate(1384,0) translate(0 -0.5)"><use data-c="7C" xlink:href="#MJX-9-TEX-N-7C"></use></g><g data-mml-node="mo" transform="translate(1939.8,0)"><use data-c="3D" xlink:href="#MJX-9-TEX-N-3D"></use></g><g data-mml-node="msqrt" transform="translate(2995.6,0)"><g transform="translate(1020,0)"><g data-mml-node="mi"><use data-c="1D453" xlink:href="#MJX-9-TEX-I-1D453"></use></g><g data-mml-node="mo" transform="translate(772.2,0)"><use data-c="22C5" xlink:href="#MJX-9-TEX-N-22C5"></use></g><g data-mml-node="mi" transform="translate(1272.4,0)"><use data-c="1D453" 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transform="translate(457,266) translate(-250 0)"><use data-c="AF" xlink:href="#MJX-9-TEX-N-AF"></use></g></g></g><g data-mml-node="mo" transform="translate(3771.3,0)"><use data-c="28" xlink:href="#MJX-9-TEX-N-28"></use></g><g data-mml-node="mi" transform="translate(4160.3,0)"><use data-c="1D456" xlink:href="#MJX-9-TEX-I-1D456"></use></g><g data-mml-node="mo" transform="translate(4505.3,0)"><use data-c="2C" xlink:href="#MJX-9-TEX-N-2C"></use></g><g data-mml-node="mi" transform="translate(4950,0)"><use data-c="1D457" xlink:href="#MJX-9-TEX-I-1D457"></use></g><g data-mml-node="mo" transform="translate(5362,0)"><use data-c="29" xlink:href="#MJX-9-TEX-N-29"></use></g><g data-mml-node="mi" transform="translate(5751,0)"><use data-c="1D453" xlink:href="#MJX-9-TEX-I-1D453"></use></g><g data-mml-node="mo" transform="translate(6301,0)"><use data-c="28" xlink:href="#MJX-9-TEX-N-28"></use></g><g data-mml-node="mi" transform="translate(6690,0)"><use data-c="1D456" xlink:href="#MJX-9-TEX-I-1D456"></use></g><g data-mml-node="mo" transform="translate(7035,0)"><use data-c="2C" xlink:href="#MJX-9-TEX-N-2C"></use></g><g data-mml-node="mi" transform="translate(7479.7,0)"><use data-c="1D457" xlink:href="#MJX-9-TEX-I-1D457"></use></g><g data-mml-node="mo" transform="translate(7891.7,0)"><use data-c="29" xlink:href="#MJX-9-TEX-N-29"></use></g></g><g data-mml-node="mo" transform="translate(0,-434.4)"><use data-c="E001" xlink:href="#MJX-9-TEX-S4-E001" transform="translate(0,1945)"></use><use data-c="23B7" xlink:href="#MJX-9-TEX-S4-23B7" transform="translate(0,-165)"></use><svg width="1056" height="1361" y="670" x="0" viewBox="0 295.5 1056 1361"><use data-c="E000" xlink:href="#MJX-9-TEX-S4-E000" transform="scale(1,3.195)"></use></svg></g><rect width="8280.7" height="60" x="1056" y="2055.6"></rect></g><g data-mml-node="mo" transform="translate(16508.2,0)"><use data-c="2E" xlink:href="#MJX-9-TEX-N-2E"></use></g></g></g></svg><mjx-assistive-mml unselectable="on" display="block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mo data-mjx-texclass="ORD" stretchy="false">|</mo><mo data-mjx-texclass="ORD" stretchy="false">|</mo><mi>f</mi><mo data-mjx-texclass="ORD" stretchy="false">|</mo><mo data-mjx-texclass="ORD" stretchy="false">|</mo><mo>=</mo><msqrt><mi>f</mi><mo>⋅</mo><mi>f</mi></msqrt><mo>=</mo><msqrt><munderover><mo data-mjx-texclass="OP">∑</mo><mrow data-mjx-texclass="ORD"><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>M</mi></munderover><munderover><mo data-mjx-texclass="OP">∑</mo><mrow data-mjx-texclass="ORD"><mi>j</mi><mo>=</mo><mn>1</mn></mrow><mi>N</mi></munderover><mrow data-mjx-texclass="ORD"><mover><mi>f</mi><mo stretchy="false">¯</mo></mover></mrow><mo stretchy="false">(</mo><mi>i</mi><mo>,</mo><mi>j</mi><mo stretchy="false">)</mo><mi>f</mi><mo stretchy="false">(</mo><mi>i</mi><mo>,</mo><mi>j</mi><mo stretchy="false">)</mo></msqrt><mo>.</mo></math></mjx-assistive-mml></mjx-container></div></div><p><span>The results are as follows:</span></p><ul><li><p><span>Norm of u (</span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="3.81ex" height="2.26ex" role="img" focusable="false" viewBox="0 -749.5 1684 999" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.564ex;"><defs><path id="MJX-68-TEX-N-7C" d="M139 -249H137Q125 -249 119 -235V251L120 737Q130 750 139 750Q152 750 159 735V-235Q151 -249 141 -249H139Z"></path><path id="MJX-68-TEX-I-1D462" d="M21 287Q21 295 30 318T55 370T99 420T158 442Q204 442 227 417T250 358Q250 340 216 246T182 105Q182 62 196 45T238 27T291 44T328 78L339 95Q341 99 377 247Q407 367 413 387T427 416Q444 431 463 431Q480 431 488 421T496 402L420 84Q419 79 419 68Q419 43 426 35T447 26Q469 29 482 57T512 145Q514 153 532 153Q551 153 551 144Q550 139 549 130T540 98T523 55T498 17T462 -8Q454 -10 438 -10Q372 -10 347 46Q345 45 336 36T318 21T296 6T267 -6T233 -11Q189 -11 155 7Q103 38 103 113Q103 170 138 262T173 379Q173 380 173 381Q173 390 173 393T169 400T158 404H154Q131 404 112 385T82 344T65 302T57 280Q55 278 41 278H27Q21 284 21 287Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mo" transform="translate(0 -0.5)"><use data-c="7C" xlink:href="#MJX-68-TEX-N-7C"></use></g><g data-mml-node="mo" transform="translate(278,0) translate(0 -0.5)"><use data-c="7C" xlink:href="#MJX-68-TEX-N-7C"></use></g><g data-mml-node="mi" transform="translate(556,0)"><use data-c="1D462" xlink:href="#MJX-68-TEX-I-1D462"></use></g><g data-mml-node="mo" transform="translate(1128,0) translate(0 -0.5)"><use data-c="7C" xlink:href="#MJX-68-TEX-N-7C"></use></g><g data-mml-node="mo" transform="translate(1406,0) translate(0 -0.5)"><use data-c="7C" xlink:href="#MJX-68-TEX-N-7C"></use></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo data-mjx-texclass="ORD" stretchy="false">|</mo><mo data-mjx-texclass="ORD" stretchy="false">|</mo><mi>u</mi><mo data-mjx-texclass="ORD" stretchy="false">|</mo><mo data-mjx-texclass="ORD" stretchy="false">|</mo></math></mjx-assistive-mml></mjx-container><script type="math/tex">||u||</script><span>): 8.660</span></p></li><li><p><span>Norm of v (</span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="3.613ex" height="2.26ex" role="img" focusable="false" viewBox="0 -749.5 1597 999" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.564ex;"><defs><path id="MJX-69-TEX-N-7C" d="M139 -249H137Q125 -249 119 -235V251L120 737Q130 750 139 750Q152 750 159 735V-235Q151 -249 141 -249H139Z"></path><path id="MJX-69-TEX-I-1D463" d="M173 380Q173 405 154 405Q130 405 104 376T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Q21 294 29 316T53 368T97 419T160 441Q202 441 225 417T249 361Q249 344 246 335Q246 329 231 291T200 202T182 113Q182 86 187 69Q200 26 250 26Q287 26 319 60T369 139T398 222T409 277Q409 300 401 317T383 343T365 361T357 383Q357 405 376 424T417 443Q436 443 451 425T467 367Q467 340 455 284T418 159T347 40T241 -11Q177 -11 139 22Q102 54 102 117Q102 148 110 181T151 298Q173 362 173 380Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mo" transform="translate(0 -0.5)"><use data-c="7C" xlink:href="#MJX-69-TEX-N-7C"></use></g><g data-mml-node="mo" transform="translate(278,0) translate(0 -0.5)"><use data-c="7C" xlink:href="#MJX-69-TEX-N-7C"></use></g><g data-mml-node="mi" transform="translate(556,0)"><use data-c="1D463" xlink:href="#MJX-69-TEX-I-1D463"></use></g><g data-mml-node="mo" transform="translate(1041,0) translate(0 -0.5)"><use data-c="7C" xlink:href="#MJX-69-TEX-N-7C"></use></g><g data-mml-node="mo" transform="translate(1319,0) translate(0 -0.5)"><use data-c="7C" xlink:href="#MJX-69-TEX-N-7C"></use></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo data-mjx-texclass="ORD" stretchy="false">|</mo><mo data-mjx-texclass="ORD" stretchy="false">|</mo><mi>v</mi><mo data-mjx-texclass="ORD" stretchy="false">|</mo><mo data-mjx-texclass="ORD" stretchy="false">|</mo></math></mjx-assistive-mml></mjx-container><script type="math/tex">||v||</script><span>): 1</span></p></li><li><p><span>Norm of w (</span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="4.136ex" height="2.26ex" role="img" focusable="false" viewBox="0 -749.5 1828 999" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.564ex;"><defs><path id="MJX-96-TEX-N-7C" d="M139 -249H137Q125 -249 119 -235V251L120 737Q130 750 139 750Q152 750 159 735V-235Q151 -249 141 -249H139Z"></path><path id="MJX-96-TEX-I-1D464" d="M580 385Q580 406 599 424T641 443Q659 443 674 425T690 368Q690 339 671 253Q656 197 644 161T609 80T554 12T482 -11Q438 -11 404 5T355 48Q354 47 352 44Q311 -11 252 -11Q226 -11 202 -5T155 14T118 53T104 116Q104 170 138 262T173 379Q173 380 173 381Q173 390 173 393T169 400T158 404H154Q131 404 112 385T82 344T65 302T57 280Q55 278 41 278H27Q21 284 21 287Q21 293 29 315T52 366T96 418T161 441Q204 441 227 416T250 358Q250 340 217 250T184 111Q184 65 205 46T258 26Q301 26 334 87L339 96V119Q339 122 339 128T340 136T341 143T342 152T345 165T348 182T354 206T362 238T373 281Q402 395 406 404Q419 431 449 431Q468 431 475 421T483 402Q483 389 454 274T422 142Q420 131 420 107V100Q420 85 423 71T442 42T487 26Q558 26 600 148Q609 171 620 213T632 273Q632 306 619 325T593 357T580 385Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mo" transform="translate(0 -0.5)"><use data-c="7C" xlink:href="#MJX-96-TEX-N-7C"></use></g><g data-mml-node="mo" transform="translate(278,0) translate(0 -0.5)"><use data-c="7C" xlink:href="#MJX-96-TEX-N-7C"></use></g><g data-mml-node="mi" transform="translate(556,0)"><use data-c="1D464" xlink:href="#MJX-96-TEX-I-1D464"></use></g><g data-mml-node="mo" transform="translate(1272,0) translate(0 -0.5)"><use data-c="7C" xlink:href="#MJX-96-TEX-N-7C"></use></g><g data-mml-node="mo" transform="translate(1550,0) translate(0 -0.5)"><use data-c="7C" xlink:href="#MJX-96-TEX-N-7C"></use></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo data-mjx-texclass="ORD" stretchy="false">|</mo><mo data-mjx-texclass="ORD" stretchy="false">|</mo><mi>w</mi><mo data-mjx-texclass="ORD" stretchy="false">|</mo><mo data-mjx-texclass="ORD" stretchy="false">|</mo></math></mjx-assistive-mml></mjx-container><script type="math/tex">||w||</script><span>): 1</span></p></li><li><p><span>Scalar product of u and v (</span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="4.026ex" height="1.027ex" role="img" focusable="false" viewBox="0 -443 1779.4 454" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.025ex;"><defs><path id="MJX-71-TEX-I-1D462" d="M21 287Q21 295 30 318T55 370T99 420T158 442Q204 442 227 417T250 358Q250 340 216 246T182 105Q182 62 196 45T238 27T291 44T328 78L339 95Q341 99 377 247Q407 367 413 387T427 416Q444 431 463 431Q480 431 488 421T496 402L420 84Q419 79 419 68Q419 43 426 35T447 26Q469 29 482 57T512 145Q514 153 532 153Q551 153 551 144Q550 139 549 130T540 98T523 55T498 17T462 -8Q454 -10 438 -10Q372 -10 347 46Q345 45 336 36T318 21T296 6T267 -6T233 -11Q189 -11 155 7Q103 38 103 113Q103 170 138 262T173 379Q173 380 173 381Q173 390 173 393T169 400T158 404H154Q131 404 112 385T82 344T65 302T57 280Q55 278 41 278H27Q21 284 21 287Z"></path><path id="MJX-71-TEX-N-B7" d="M78 250Q78 274 95 292T138 310Q162 310 180 294T199 251Q199 226 182 208T139 190T96 207T78 250Z"></path><path id="MJX-71-TEX-I-1D463" d="M173 380Q173 405 154 405Q130 405 104 376T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Q21 294 29 316T53 368T97 419T160 441Q202 441 225 417T249 361Q249 344 246 335Q246 329 231 291T200 202T182 113Q182 86 187 69Q200 26 250 26Q287 26 319 60T369 139T398 222T409 277Q409 300 401 317T383 343T365 361T357 383Q357 405 376 424T417 443Q436 443 451 425T467 367Q467 340 455 284T418 159T347 40T241 -11Q177 -11 139 22Q102 54 102 117Q102 148 110 181T151 298Q173 362 173 380Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><use data-c="1D462" xlink:href="#MJX-71-TEX-I-1D462"></use></g><g data-mml-node="mo" transform="translate(794.2,0)"><use data-c="B7" xlink:href="#MJX-71-TEX-N-B7"></use></g><g data-mml-node="mi" transform="translate(1294.4,0)"><use data-c="1D463" xlink:href="#MJX-71-TEX-I-1D463"></use></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>u</mi><mo>·</mo><mi>v</mi></math></mjx-assistive-mml></mjx-container><script type="math/tex">u · v</script><span>)= 7.50</span></p></li><li><p><span>Scalar product of u and w </span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="6.309ex" height="2.262ex" role="img" focusable="false" viewBox="0 -750 2788.4 1000" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.566ex;"><defs><path id="MJX-72-TEX-N-28" d="M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z"></path><path id="MJX-72-TEX-I-1D462" d="M21 287Q21 295 30 318T55 370T99 420T158 442Q204 442 227 417T250 358Q250 340 216 246T182 105Q182 62 196 45T238 27T291 44T328 78L339 95Q341 99 377 247Q407 367 413 387T427 416Q444 431 463 431Q480 431 488 421T496 402L420 84Q419 79 419 68Q419 43 426 35T447 26Q469 29 482 57T512 145Q514 153 532 153Q551 153 551 144Q550 139 549 130T540 98T523 55T498 17T462 -8Q454 -10 438 -10Q372 -10 347 46Q345 45 336 36T318 21T296 6T267 -6T233 -11Q189 -11 155 7Q103 38 103 113Q103 170 138 262T173 379Q173 380 173 381Q173 390 173 393T169 400T158 404H154Q131 404 112 385T82 344T65 302T57 280Q55 278 41 278H27Q21 284 21 287Z"></path><path id="MJX-72-TEX-N-B7" d="M78 250Q78 274 95 292T138 310Q162 310 180 294T199 251Q199 226 182 208T139 190T96 207T78 250Z"></path><path id="MJX-72-TEX-I-1D464" d="M580 385Q580 406 599 424T641 443Q659 443 674 425T690 368Q690 339 671 253Q656 197 644 161T609 80T554 12T482 -11Q438 -11 404 5T355 48Q354 47 352 44Q311 -11 252 -11Q226 -11 202 -5T155 14T118 53T104 116Q104 170 138 262T173 379Q173 380 173 381Q173 390 173 393T169 400T158 404H154Q131 404 112 385T82 344T65 302T57 280Q55 278 41 278H27Q21 284 21 287Q21 293 29 315T52 366T96 418T161 441Q204 441 227 416T250 358Q250 340 217 250T184 111Q184 65 205 46T258 26Q301 26 334 87L339 96V119Q339 122 339 128T340 136T341 143T342 152T345 165T348 182T354 206T362 238T373 281Q402 395 406 404Q419 431 449 431Q468 431 475 421T483 402Q483 389 454 274T422 142Q420 131 420 107V100Q420 85 423 71T442 42T487 26Q558 26 600 148Q609 171 620 213T632 273Q632 306 619 325T593 357T580 385Z"></path><path id="MJX-72-TEX-N-29" d="M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mo"><use data-c="28" xlink:href="#MJX-72-TEX-N-28"></use></g><g data-mml-node="mi" transform="translate(389,0)"><use data-c="1D462" xlink:href="#MJX-72-TEX-I-1D462"></use></g><g data-mml-node="mo" transform="translate(1183.2,0)"><use data-c="B7" xlink:href="#MJX-72-TEX-N-B7"></use></g><g data-mml-node="mi" transform="translate(1683.4,0)"><use data-c="1D464" xlink:href="#MJX-72-TEX-I-1D464"></use></g><g data-mml-node="mo" transform="translate(2399.4,0)"><use data-c="29" xlink:href="#MJX-72-TEX-N-29"></use></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">(</mo><mi>u</mi><mo>·</mo><mi>w</mi><mo stretchy="false">)</mo></math></mjx-assistive-mml></mjx-container><script type="math/tex">(u · w)</script><span>= -2.50</span></p></li><li><p><span>Scalar product of v and w </span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="6.112ex" height="2.262ex" role="img" focusable="false" viewBox="0 -750 2701.4 1000" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.566ex;"><defs><path id="MJX-73-TEX-N-28" d="M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z"></path><path id="MJX-73-TEX-I-1D463" d="M173 380Q173 405 154 405Q130 405 104 376T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Q21 294 29 316T53 368T97 419T160 441Q202 441 225 417T249 361Q249 344 246 335Q246 329 231 291T200 202T182 113Q182 86 187 69Q200 26 250 26Q287 26 319 60T369 139T398 222T409 277Q409 300 401 317T383 343T365 361T357 383Q357 405 376 424T417 443Q436 443 451 425T467 367Q467 340 455 284T418 159T347 40T241 -11Q177 -11 139 22Q102 54 102 117Q102 148 110 181T151 298Q173 362 173 380Z"></path><path id="MJX-73-TEX-N-B7" d="M78 250Q78 274 95 292T138 310Q162 310 180 294T199 251Q199 226 182 208T139 190T96 207T78 250Z"></path><path id="MJX-73-TEX-I-1D464" d="M580 385Q580 406 599 424T641 443Q659 443 674 425T690 368Q690 339 671 253Q656 197 644 161T609 80T554 12T482 -11Q438 -11 404 5T355 48Q354 47 352 44Q311 -11 252 -11Q226 -11 202 -5T155 14T118 53T104 116Q104 170 138 262T173 379Q173 380 173 381Q173 390 173 393T169 400T158 404H154Q131 404 112 385T82 344T65 302T57 280Q55 278 41 278H27Q21 284 21 287Q21 293 29 315T52 366T96 418T161 441Q204 441 227 416T250 358Q250 340 217 250T184 111Q184 65 205 46T258 26Q301 26 334 87L339 96V119Q339 122 339 128T340 136T341 143T342 152T345 165T348 182T354 206T362 238T373 281Q402 395 406 404Q419 431 449 431Q468 431 475 421T483 402Q483 389 454 274T422 142Q420 131 420 107V100Q420 85 423 71T442 42T487 26Q558 26 600 148Q609 171 620 213T632 273Q632 306 619 325T593 357T580 385Z"></path><path id="MJX-73-TEX-N-29" d="M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mo"><use data-c="28" xlink:href="#MJX-73-TEX-N-28"></use></g><g data-mml-node="mi" transform="translate(389,0)"><use data-c="1D463" xlink:href="#MJX-73-TEX-I-1D463"></use></g><g data-mml-node="mo" transform="translate(1096.2,0)"><use data-c="B7" xlink:href="#MJX-73-TEX-N-B7"></use></g><g data-mml-node="mi" transform="translate(1596.4,0)"><use data-c="1D464" xlink:href="#MJX-73-TEX-I-1D464"></use></g><g data-mml-node="mo" transform="translate(2312.4,0)"><use data-c="29" xlink:href="#MJX-73-TEX-N-29"></use></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">(</mo><mi>v</mi><mo>·</mo><mi>w</mi><mo stretchy="false">)</mo></math></mjx-assistive-mml></mjx-container><script type="math/tex">(v · w)</script><span>= 0</span></p></li><li><p><span>Matrices {v, w} are orthonormal because their scalar product </span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="6.892ex" height="2.262ex" role="img" focusable="false" viewBox="0 -750 3046.4 1000" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.566ex;"><defs><path id="MJX-74-TEX-N-28" d="M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z"></path><path id="MJX-74-TEX-I-1D463" d="M173 380Q173 405 154 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displaystyle="true"><mtr><mtd><mi>p</mi><mi>r</mi><mi>o</mi><mi>j</mi><mi>e</mi><mi>c</mi><mi>t</mi><mi>i</mi><mi>o</mi><mi>n</mi><mo>=</mo><mn>7.5</mn><mi>v</mi><mo>+</mo><mo>−</mo><mn>2.5</mn><mi>w</mi><mo>=</mo><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">[</mo><mtable columnspacing="1em" rowspacing="4pt"><mtr><mtd><mn>5</mn></mtd><mtd><mo>−</mo><mn>5</mn></mtd></mtr><mtr><mtd><mo>−</mo><mn>2.5</mn></mtd><mtd><mn>2.5</mn></mtd></mtr></mtable><mo data-mjx-texclass="CLOSE">]</mo></mrow></mtd></mtr></mtable></math></mjx-assistive-mml></mjx-container></div></div></li><li><p><span>The projection is the closest to u. The resulting projection may not be exactly equal to u, and the error is 12.5, but it will be the best approximation of u within the subspace.</span></p><p><span>  </span></p></li></ul><h2 id='7--image-compression'><span>7.  Image compression</span></h2><ul><li><p><span>I use MATLAB to calculate the dot production of two image, to test if any of them are orthonormal.</span>
<span>First, I test their L2 norms to see if their norms are 1.</span></p><pre class="md-fences md-end-block md-fences-with-lineno ty-contain-cm modeLoaded" spellcheck="false" lang="matlab"><div class="CodeMirror cm-s-inner cm-s-null-scroll CodeMirror-wrap" lang="matlab"><div style="overflow: hidden; position: relative; width: 3px; height: 0px; top: 9.51875px; left: 44px;"><textarea autocorrect="off" autocapitalize="off" spellcheck="false" tabindex="0" style="position: absolute; bottom: -1em; padding: 0px; width: 1000px; height: 1em; outline: none;"></textarea></div><div class="CodeMirror-scrollbar-filler" cm-not-content="true"></div><div class="CodeMirror-gutter-filler" cm-not-content="true"></div><div class="CodeMirror-scroll" tabindex="-1"><div class="CodeMirror-sizer" style="margin-left: 36px; margin-bottom: 0px; border-right-width: 0px; padding-right: 0px; padding-bottom: 0px;"><div style="position: relative; top: 0px;"><div class="CodeMirror-lines" role="presentation"><div role="presentation" style="position: relative; outline: none;"><div class="CodeMirror-measure"><div class="CodeMirror-linenumber CodeMirror-gutter-elt"><div>10</div></div></div><div class="CodeMirror-measure"></div><div style="position: relative; z-index: 1;"></div><div class="CodeMirror-code" role="presentation" style=""><div class="CodeMirror-activeline" style="position: relative;"><div class="CodeMirror-activeline-background CodeMirror-linebackground"></div><div class="CodeMirror-gutter-background CodeMirror-activeline-gutter" style="left: -36px; width: 36px;"></div><div class="CodeMirror-gutter-wrapper CodeMirror-activeline-gutter" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt CodeMirror-linenumber-show" style="left: 0px; width: 27px;">1</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-builtin">clear</span>;</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">2</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-variable">phi_1</span> = <span class="cm-number">1</span><span class="cm-operator">/</span><span class="cm-number">3</span> <span class="cm-operator">*</span> [<span class="cm-number">1</span> <span class="cm-number">0</span> <span class="cm-number">-1</span>; <span class="cm-number">1</span> <span class="cm-number">0</span> <span class="cm-number">-1</span>; <span class="cm-number">0</span> <span class="cm-number">0</span> <span class="cm-number">0</span>; <span class="cm-number">0</span> <span class="cm-number">0</span> <span class="cm-number">0</span>];</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">3</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-variable">phi_2</span> = <span class="cm-number">1</span><span class="cm-operator">/</span><span class="cm-number">2</span> <span class="cm-operator">*</span> [<span class="cm-number">1</span> <span class="cm-number">1</span> <span class="cm-number">1</span>; <span class="cm-number">1</span> <span class="cm-number">0</span> <span class="cm-number">1</span>; <span class="cm-number">-1</span> <span class="cm-number">-1</span> <span class="cm-number">-1</span>; <span class="cm-number">0</span> <span class="cm-number">-1</span> <span class="cm-number">0</span>];</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">4</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-variable">phi_3</span> = <span class="cm-number">1</span><span class="cm-operator">/</span><span class="cm-number">2</span> <span class="cm-operator">*</span> [<span class="cm-number">0</span> <span class="cm-number">1</span> <span class="cm-number">0</span>; <span class="cm-number">1</span> <span class="cm-number">1</span> <span class="cm-number">1</span>; <span class="cm-number">1</span> <span class="cm-number">0</span> <span class="cm-number">1</span>; <span class="cm-number">1</span> <span class="cm-number">1</span> <span class="cm-number">1</span>];</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">5</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-variable">phi_4</span> = <span class="cm-number">1</span><span class="cm-operator">/</span><span class="cm-number">3</span> <span class="cm-operator">*</span> [<span class="cm-number">0</span> <span class="cm-number">0</span> <span class="cm-number">0</span>; <span class="cm-number">0</span> <span class="cm-number">0</span> <span class="cm-number">0</span>; <span class="cm-number">1</span> <span class="cm-number">0</span> <span class="cm-number">-1</span>; <span class="cm-number">1</span> <span class="cm-number">0</span> <span class="cm-number">-1</span>];</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">6</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span cm-text="" cm-zwsp="">
</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">7</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-variable">norm_1</span> = <span class="cm-builtin">norm</span>(<span class="cm-variable">phi_1</span>, <span class="cm-string">'fro'</span>);</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">8</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-variable">norm_2</span> = <span class="cm-builtin">norm</span>(<span class="cm-variable">phi_2</span>, <span class="cm-string">'fro'</span>);</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">9</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-variable">norm_3</span> = <span class="cm-builtin">norm</span>(<span class="cm-variable">phi_3</span>, <span class="cm-string">'fro'</span>);</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt CodeMirror-linenumber-show" style="left: 0px; width: 27px;">10</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-variable">norm_4</span> = <span class="cm-builtin">norm</span>(<span class="cm-variable">phi_4</span>, <span class="cm-string">'fro'</span>);</span></pre></div></div></div></div></div></div><div style="position: absolute; height: 0px; width: 1px; border-bottom: 0px solid transparent; top: 230px;"></div><div class="CodeMirror-gutters" style="height: 230px; left: 0px;"><div class="CodeMirror-gutter CodeMirror-linenumbers" style="width: 35px;"></div></div></div></div></pre><p><span>and the results are:</span></p><p><img src="C:\Users\30379\AppData\Roaming\Typora\typora-user-images\image-20230905165723484.png" referrerpolicy="no-referrer" alt="image-20230905165723484"></p><p><span>Then I iterate through all four images, testing them pairwise to determine if they are orthogonal.</span></p><pre class="md-fences md-end-block md-fences-with-lineno ty-contain-cm modeLoaded" spellcheck="false" lang="matlab" style="break-inside: unset;"><div class="CodeMirror cm-s-inner cm-s-null-scroll CodeMirror-wrap" lang="matlab"><div style="overflow: hidden; position: relative; width: 3px; height: 0px; top: 9.51875px; left: 44px;"><textarea autocorrect="off" autocapitalize="off" spellcheck="false" tabindex="0" style="position: absolute; bottom: -1em; padding: 0px; width: 1000px; height: 1em; outline: none;"></textarea></div><div class="CodeMirror-scrollbar-filler" cm-not-content="true"></div><div class="CodeMirror-gutter-filler" cm-not-content="true"></div><div class="CodeMirror-scroll" tabindex="-1"><div class="CodeMirror-sizer" style="margin-left: 36px; margin-bottom: 0px; border-right-width: 0px; padding-right: 0px; padding-bottom: 0px;"><div style="position: relative; top: 0px;"><div class="CodeMirror-lines" role="presentation"><div role="presentation" style="position: relative; outline: none;"><div class="CodeMirror-measure"><div class="CodeMirror-linenumber CodeMirror-gutter-elt"><div>14</div></div></div><div class="CodeMirror-measure"></div><div style="position: relative; z-index: 1;"></div><div class="CodeMirror-code" role="presentation" style=""><div class="CodeMirror-activeline" style="position: relative;"><div class="CodeMirror-activeline-background CodeMirror-linebackground"></div><div class="CodeMirror-gutter-background CodeMirror-activeline-gutter" style="left: -36px; width: 36px;"></div><div class="CodeMirror-gutter-wrapper CodeMirror-activeline-gutter" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt CodeMirror-linenumber-show" style="left: 0px; width: 27px;">1</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-variable">phi</span> = {<span class="cm-variable">phi_1</span>; <span class="cm-variable">phi_2</span>; <span class="cm-variable">phi_3</span>; <span class="cm-variable">phi_4</span>};</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">2</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span cm-text="" cm-zwsp="">
</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">3</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-variable">num</span> = <span class="cm-variable">numel</span>(<span class="cm-variable">phi</span>);</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">4</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span cm-text="" cm-zwsp="">
</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">5</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-variable">scalar_products</span> = <span class="cm-builtin">zeros</span>(<span class="cm-variable">num</span>, <span class="cm-variable">num</span>);</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">6</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span cm-text="" cm-zwsp="">
</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">7</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-keyword">for</span> <span class="cm-variable">i</span> = <span class="cm-number">1</span>:<span class="cm-variable">num</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">8</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp;<span class="cm-keyword">for</span> <span class="cm-variable">j</span> = <span class="cm-number">1</span>:<span class="cm-variable">num</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">9</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp; &nbsp; &nbsp;<span class="cm-keyword">if</span> <span class="cm-variable">i</span> <span class="cm-operator">==</span> <span class="cm-variable">j</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt CodeMirror-linenumber-show" style="left: 0px; width: 27px;">10</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;<span class="cm-keyword">continue</span>; </span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">11</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp; &nbsp; &nbsp;<span class="cm-keyword">end</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">12</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp; &nbsp; &nbsp;<span class="cm-variable">scalar_products</span>(<span class="cm-variable">i</span>, <span class="cm-variable">j</span>) = <span class="cm-variable">dot</span>(<span class="cm-variable">phi</span>{<span class="cm-variable">i</span>}(:), <span class="cm-variable">phi</span>{<span class="cm-variable">j</span>}(:));</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">13</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp;<span class="cm-keyword">end</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt CodeMirror-linenumber-show" style="left: 0px; width: 27px;">14</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-keyword">end</span></span></pre></div></div></div></div></div></div><div style="position: absolute; height: 0px; width: 1px; border-bottom: 0px solid transparent; top: 323px;"></div><div class="CodeMirror-gutters" style="height: 323px; left: 0px;"><div class="CodeMirror-gutter CodeMirror-linenumbers" style="width: 35px;"></div></div></div></div></pre><p><span> The results stored in the matrix below show that these images are orthogon and also orthonormal since their L2 norms are 1.</span></p><div contenteditable="false" spellcheck="false" class="mathjax-block md-end-block md-math-block md-rawblock" id="mathjax-n115" cid="n115" mdtype="math_block" data-math-tag-before="0" data-math-tag-after="0" data-math-labels="[]"><div class="md-rawblock-container md-math-container" tabindex="-1"><mjx-container class="MathJax" jax="SVG" display="true" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="31.565ex" height="11.765ex" role="img" focusable="false" viewBox="0 -2850 13951.6 5200" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -5.317ex;"><defs><path id="MJX-11-TEX-I-1D460" d="M131 289Q131 321 147 354T203 415T300 442Q362 442 390 415T419 355Q419 323 402 308T364 292Q351 292 340 300T328 326Q328 342 337 354T354 372T367 378Q368 378 368 379Q368 382 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xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>f</mi><mi>a</mi></msub><mo>=</mo><msub><mi>x</mi><mn>1</mn></msub><msub><mi>ϕ</mi><mn>1</mn></msub><mo>+</mo><msub><mi>x</mi><mn>2</mn></msub><msub><mi>ϕ</mi><mn>2</mn></msub><mo>+</mo><msub><mi>x</mi><mn>3</mn></msub><msub><mi>ϕ</mi><mn>3</mn></msub><mo>+</mo><msub><mi>x</mi><mn>4</mn></msub><msub><mi>ϕ</mi><mn>4</mn></msub></math></mjx-assistive-mml></mjx-container><script type="math/tex">f_a = x_1\phi_1 + x_2\phi_2 + x_3\phi_3 + x_4\phi_4</script><span>, we can view this problem as finding a linear combination of these four images.</span></p><p><span>So we are actually solving x based on the equation </span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="6.979ex" height="1.805ex" role="img" focusable="false" viewBox="0 -716 3084.6 798" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.186ex;"><defs><path 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display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">Φ</mi><mi>x</mi><mo>=</mo><mi>f</mi></math></mjx-assistive-mml></mjx-container><script type="math/tex">\Phi x = f</script><span>. </span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="1.633ex" height="1.545ex" role="img" focusable="false" viewBox="0 -683 722 683" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: 0px;"><defs><path id="MJX-79-TEX-N-3A6" d="M312 622Q310 623 307 625T303 629T297 631T286 634T270 635T246 636T211 637H184V683H196Q220 680 361 680T526 683H538V637H511Q468 637 447 635T422 631T411 622V533L425 531Q525 519 595 466T665 342Q665 301 642 267T583 209T506 172T425 152L411 150V61Q417 55 421 53T447 48T511 46H538V0H526Q502 3 361 3T196 0H184V46H211Q231 46 245 46T270 47T286 48T297 51T303 54T307 57T312 61V150H310Q309 151 289 153T232 166T160 195Q149 201 136 210T103 238T69 284T56 342Q56 414 128 467T294 530Q309 532 310 533H312V622ZM170 342Q170 207 307 188H312V495H309Q301 495 282 491T231 469T186 423Q170 389 170 342ZM415 188Q487 199 519 236T551 342Q551 384 539 414T507 459T470 481T434 491T415 495H410V188H415Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><use data-c="3A6" xlink:href="#MJX-79-TEX-N-3A6"></use></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">Φ</mi></math></mjx-assistive-mml></mjx-container><script type="math/tex">\Phi</script><span> is a matrix that we construct with </span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="12.362ex" height="2.034ex" role="img" focusable="false" viewBox="0 -694 5464.2 899" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: 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xlink:href="#MJX-80-TEX-I-1D719"></use></g><g data-mml-node="mn" transform="translate(629,-150) scale(0.707)"><use data-c="31" xlink:href="#MJX-80-TEX-N-31"></use></g></g><g data-mml-node="mo" transform="translate(1032.6,0)"><use data-c="2C" xlink:href="#MJX-80-TEX-N-2C"></use></g><g data-mml-node="msub" transform="translate(1477.2,0)"><g data-mml-node="mi"><use data-c="1D719" xlink:href="#MJX-80-TEX-I-1D719"></use></g><g data-mml-node="mn" transform="translate(629,-150) scale(0.707)"><use data-c="32" xlink:href="#MJX-80-TEX-N-32"></use></g></g><g data-mml-node="mo" transform="translate(2509.8,0)"><use data-c="2C" xlink:href="#MJX-80-TEX-N-2C"></use></g><g data-mml-node="msub" transform="translate(2954.4,0)"><g data-mml-node="mi"><use data-c="1D719" xlink:href="#MJX-80-TEX-I-1D719"></use></g><g data-mml-node="mn" transform="translate(629,-150) scale(0.707)"><use data-c="33" xlink:href="#MJX-80-TEX-N-33"></use></g></g><g data-mml-node="mo" transform="translate(3987,0)"><use data-c="2C" xlink:href="#MJX-80-TEX-N-2C"></use></g><g data-mml-node="msub" transform="translate(4431.7,0)"><g data-mml-node="mi"><use data-c="1D719" xlink:href="#MJX-80-TEX-I-1D719"></use></g><g data-mml-node="mn" transform="translate(629,-150) scale(0.707)"><use data-c="34" xlink:href="#MJX-80-TEX-N-34"></use></g></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>ϕ</mi><mn>1</mn></msub><mo>,</mo><msub><mi>ϕ</mi><mn>2</mn></msub><mo>,</mo><msub><mi>ϕ</mi><mn>3</mn></msub><mo>,</mo><msub><mi>ϕ</mi><mn>4</mn></msub></math></mjx-assistive-mml></mjx-container><script type="math/tex">\phi_1,\phi_2,\phi_3,\phi_4</script><span> as the columns, x is the linear combination of these four images.</span></p><p><span>Since we only have 4 basis vectors and the image f is of dimension </span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="10.308ex" height="1.717ex" role="img" focusable="false" viewBox="0 -677 4556 759" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.186ex;"><defs><path id="MJX-81-TEX-N-33" d="M127 463Q100 463 85 480T69 524Q69 579 117 622T233 665Q268 665 277 664Q351 652 390 611T430 522Q430 470 396 421T302 350L299 348Q299 347 308 345T337 336T375 315Q457 262 457 175Q457 96 395 37T238 -22Q158 -22 100 21T42 130Q42 158 60 175T105 193Q133 193 151 175T169 130Q169 119 166 110T159 94T148 82T136 74T126 70T118 67L114 66Q165 21 238 21Q293 21 321 74Q338 107 338 175V195Q338 290 274 322Q259 328 213 329L171 330L168 332Q166 335 166 348Q166 366 174 366Q202 366 232 371Q266 376 294 413T322 525V533Q322 590 287 612Q265 626 240 626Q208 626 181 615T143 592T132 580H135Q138 579 143 578T153 573T165 566T175 555T183 540T186 520Q186 498 172 481T127 463Z"></path><path id="MJX-81-TEX-N-D7" d="M630 29Q630 9 609 9Q604 9 587 25T493 118L389 222L284 117Q178 13 175 11Q171 9 168 9Q160 9 154 15T147 29Q147 36 161 51T255 146L359 250L255 354Q174 435 161 449T147 471Q147 480 153 485T168 490Q173 490 175 489Q178 487 284 383L389 278L493 382Q570 459 587 475T609 491Q630 491 630 471Q630 464 620 453T522 355L418 250L522 145Q606 61 618 48T630 29Z"></path><path id="MJX-81-TEX-N-34" d="M462 0Q444 3 333 3Q217 3 199 0H190V46H221Q241 46 248 46T265 48T279 53T286 61Q287 63 287 115V165H28V211L179 442Q332 674 334 675Q336 677 355 677H373L379 671V211H471V165H379V114Q379 73 379 66T385 54Q393 47 442 46H471V0H462ZM293 211V545L74 212L183 211H293Z"></path><path id="MJX-81-TEX-N-3D" d="M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z"></path><path id="MJX-81-TEX-N-31" d="M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z"></path><path id="MJX-81-TEX-N-32" d="M109 429Q82 429 66 447T50 491Q50 562 103 614T235 666Q326 666 387 610T449 465Q449 422 429 383T381 315T301 241Q265 210 201 149L142 93L218 92Q375 92 385 97Q392 99 409 186V189H449V186Q448 183 436 95T421 3V0H50V19V31Q50 38 56 46T86 81Q115 113 136 137Q145 147 170 174T204 211T233 244T261 278T284 308T305 340T320 369T333 401T340 431T343 464Q343 527 309 573T212 619Q179 619 154 602T119 569T109 550Q109 549 114 549Q132 549 151 535T170 489Q170 464 154 447T109 429Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mn"><use data-c="33" xlink:href="#MJX-81-TEX-N-33"></use></g><g data-mml-node="mo" transform="translate(722.2,0)"><use data-c="D7" xlink:href="#MJX-81-TEX-N-D7"></use></g><g data-mml-node="mn" transform="translate(1722.4,0)"><use data-c="34" xlink:href="#MJX-81-TEX-N-34"></use></g><g data-mml-node="mo" transform="translate(2500.2,0)"><use data-c="3D" xlink:href="#MJX-81-TEX-N-3D"></use></g><g data-mml-node="mn" transform="translate(3556,0)"><use data-c="31" xlink:href="#MJX-81-TEX-N-31"></use><use data-c="32" xlink:href="#MJX-81-TEX-N-32" transform="translate(500,0)"></use></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn><mo>×</mo><mn>4</mn><mo>=</mo><mn>12</mn></math></mjx-assistive-mml></mjx-container><script type="math/tex">3\times4=12</script><span>, we cannot find a solution. However, we can find a projection of f onto the subspace spanned by these 4 basis vectors using the equation </span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="13ex" height="2.09ex" role="img" focusable="false" viewBox="0 -841.7 5746.2 923.7" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.186ex;"><defs><path id="MJX-82-TEX-I-1D434" d="M208 74Q208 50 254 46Q272 46 272 35Q272 34 270 22Q267 8 264 4T251 0Q249 0 239 0T205 1T141 2Q70 2 50 0H42Q35 7 35 11Q37 38 48 46H62Q132 49 164 96Q170 102 345 401T523 704Q530 716 547 716H555H572Q578 707 578 706L606 383Q634 60 636 57Q641 46 701 46Q726 46 726 36Q726 34 723 22Q720 7 718 4T704 0Q701 0 690 0T651 1T578 2Q484 2 455 0H443Q437 6 437 9T439 27Q443 40 445 43L449 46H469Q523 49 533 63L521 213H283L249 155Q208 86 208 74ZM516 260Q516 271 504 416T490 562L463 519Q447 492 400 412L310 260L413 259Q516 259 516 260Z"></path><path id="MJX-82-TEX-I-1D447" d="M40 437Q21 437 21 445Q21 450 37 501T71 602L88 651Q93 669 101 677H569H659Q691 677 697 676T704 667Q704 661 687 553T668 444Q668 437 649 437Q640 437 637 437T631 442L629 445Q629 451 635 490T641 551Q641 586 628 604T573 629Q568 630 515 631Q469 631 457 630T439 622Q438 621 368 343T298 60Q298 48 386 46Q418 46 427 45T436 36Q436 31 433 22Q429 4 424 1L422 0Q419 0 415 0Q410 0 363 1T228 2Q99 2 64 0H49Q43 6 43 9T45 27Q49 40 55 46H83H94Q174 46 189 55Q190 56 191 56Q196 59 201 76T241 233Q258 301 269 344Q339 619 339 625Q339 630 310 630H279Q212 630 191 624Q146 614 121 583T67 467Q60 445 57 441T43 437H40Z"></path><path id="MJX-82-TEX-I-1D465" d="M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z"></path><path id="MJX-82-TEX-N-5E" d="M112 560L249 694L257 686Q387 562 387 560L361 531Q359 532 303 581L250 627L195 580Q182 569 169 557T148 538L140 532Q138 530 125 546L112 560Z"></path><path id="MJX-82-TEX-N-3D" d="M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z"></path><path id="MJX-82-TEX-I-1D44F" d="M73 647Q73 657 77 670T89 683Q90 683 161 688T234 694Q246 694 246 685T212 542Q204 508 195 472T180 418L176 399Q176 396 182 402Q231 442 283 442Q345 442 383 396T422 280Q422 169 343 79T173 -11Q123 -11 82 27T40 150V159Q40 180 48 217T97 414Q147 611 147 623T109 637Q104 637 101 637H96Q86 637 83 637T76 640T73 647ZM336 325V331Q336 405 275 405Q258 405 240 397T207 376T181 352T163 330L157 322L136 236Q114 150 114 114Q114 66 138 42Q154 26 178 26Q211 26 245 58Q270 81 285 114T318 219Q336 291 336 325Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="msup"><g data-mml-node="mi"><use data-c="1D434" xlink:href="#MJX-82-TEX-I-1D434"></use></g><g data-mml-node="mi" transform="translate(783,363) scale(0.707)"><use data-c="1D447" xlink:href="#MJX-82-TEX-I-1D447"></use></g></g><g data-mml-node="mi" transform="translate(1330.8,0)"><use data-c="1D434" xlink:href="#MJX-82-TEX-I-1D434"></use></g><g data-mml-node="TeXAtom" data-mjx-texclass="ORD" transform="translate(2080.8,0)"><g data-mml-node="mover"><g data-mml-node="mi"><use data-c="1D465" xlink:href="#MJX-82-TEX-I-1D465"></use></g><g data-mml-node="mo" transform="translate(313.8,16) translate(-250 0)"><use data-c="5E" xlink:href="#MJX-82-TEX-N-5E"></use></g></g></g><g data-mml-node="mo" transform="translate(2930.6,0)"><use data-c="3D" xlink:href="#MJX-82-TEX-N-3D"></use></g><g data-mml-node="msup" transform="translate(3986.4,0)"><g data-mml-node="mi"><use data-c="1D434" xlink:href="#MJX-82-TEX-I-1D434"></use></g><g data-mml-node="mi" transform="translate(783,363) scale(0.707)"><use data-c="1D447" xlink:href="#MJX-82-TEX-I-1D447"></use></g></g><g data-mml-node="mi" transform="translate(5317.2,0)"><use data-c="1D44F" xlink:href="#MJX-82-TEX-I-1D44F"></use></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>A</mi><mi>T</mi></msup><mi>A</mi><mrow data-mjx-texclass="ORD"><mover><mi>x</mi><mo stretchy="false">^</mo></mover></mrow><mo>=</mo><msup><mi>A</mi><mi>T</mi></msup><mi>b</mi></math></mjx-assistive-mml></mjx-container><script type="math/tex">A^TA\hat{x} = A^Tb</script><span>, or in this case, </span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="13.084ex" height="2.368ex" role="img" focusable="false" viewBox="0 -841.7 5783.2 1046.7" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.464ex;"><defs><path id="MJX-83-TEX-N-3A6" d="M312 622Q310 623 307 625T303 629T297 631T286 634T270 635T246 636T211 637H184V683H196Q220 680 361 680T526 683H538V637H511Q468 637 447 635T422 631T411 622V533L425 531Q525 519 595 466T665 342Q665 301 642 267T583 209T506 172T425 152L411 150V61Q417 55 421 53T447 48T511 46H538V0H526Q502 3 361 3T196 0H184V46H211Q231 46 245 46T270 47T286 48T297 51T303 54T307 57T312 61V150H310Q309 151 289 153T232 166T160 195Q149 201 136 210T103 238T69 284T56 342Q56 414 128 467T294 530Q309 532 310 533H312V622ZM170 342Q170 207 307 188H312V495H309Q301 495 282 491T231 469T186 423Q170 389 170 342ZM415 188Q487 199 519 236T551 342Q551 384 539 414T507 459T470 481T434 491T415 495H410V188H415Z"></path><path id="MJX-83-TEX-I-1D447" d="M40 437Q21 437 21 445Q21 450 37 501T71 602L88 651Q93 669 101 677H569H659Q691 677 697 676T704 667Q704 661 687 553T668 444Q668 437 649 437Q640 437 637 437T631 442L629 445Q629 451 635 490T641 551Q641 586 628 604T573 629Q568 630 515 631Q469 631 457 630T439 622Q438 621 368 343T298 60Q298 48 386 46Q418 46 427 45T436 36Q436 31 433 22Q429 4 424 1L422 0Q419 0 415 0Q410 0 363 1T228 2Q99 2 64 0H49Q43 6 43 9T45 27Q49 40 55 46H83H94Q174 46 189 55Q190 56 191 56Q196 59 201 76T241 233Q258 301 269 344Q339 619 339 625Q339 630 310 630H279Q212 630 191 624Q146 614 121 583T67 467Q60 445 57 441T43 437H40Z"></path><path id="MJX-83-TEX-I-1D465" d="M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z"></path><path id="MJX-83-TEX-N-5E" d="M112 560L249 694L257 686Q387 562 387 560L361 531Q359 532 303 581L250 627L195 580Q182 569 169 557T148 538L140 532Q138 530 125 546L112 560Z"></path><path id="MJX-83-TEX-N-3D" d="M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z"></path><path id="MJX-83-TEX-I-1D453" d="M118 -162Q120 -162 124 -164T135 -167T147 -168Q160 -168 171 -155T187 -126Q197 -99 221 27T267 267T289 382V385H242Q195 385 192 387Q188 390 188 397L195 425Q197 430 203 430T250 431Q298 431 298 432Q298 434 307 482T319 540Q356 705 465 705Q502 703 526 683T550 630Q550 594 529 578T487 561Q443 561 443 603Q443 622 454 636T478 657L487 662Q471 668 457 668Q445 668 434 658T419 630Q412 601 403 552T387 469T380 433Q380 431 435 431Q480 431 487 430T498 424Q499 420 496 407T491 391Q489 386 482 386T428 385H372L349 263Q301 15 282 -47Q255 -132 212 -173Q175 -205 139 -205Q107 -205 81 -186T55 -132Q55 -95 76 -78T118 -61Q162 -61 162 -103Q162 -122 151 -136T127 -157L118 -162Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="msup"><g data-mml-node="mi"><use data-c="3A6" xlink:href="#MJX-83-TEX-N-3A6"></use></g><g data-mml-node="mi" transform="translate(755,363) scale(0.707)"><use data-c="1D447" xlink:href="#MJX-83-TEX-I-1D447"></use></g></g><g data-mml-node="mi" transform="translate(1302.8,0)"><use data-c="3A6" xlink:href="#MJX-83-TEX-N-3A6"></use></g><g data-mml-node="TeXAtom" data-mjx-texclass="ORD" transform="translate(2024.8,0)"><g data-mml-node="mover"><g data-mml-node="mi"><use data-c="1D465" xlink:href="#MJX-83-TEX-I-1D465"></use></g><g data-mml-node="mo" transform="translate(313.8,16) translate(-250 0)"><use data-c="5E" xlink:href="#MJX-83-TEX-N-5E"></use></g></g></g><g data-mml-node="mo" transform="translate(2874.6,0)"><use data-c="3D" xlink:href="#MJX-83-TEX-N-3D"></use></g><g data-mml-node="msup" transform="translate(3930.4,0)"><g data-mml-node="mi"><use data-c="3A6" xlink:href="#MJX-83-TEX-N-3A6"></use></g><g data-mml-node="mi" transform="translate(755,363) scale(0.707)"><use data-c="1D447" xlink:href="#MJX-83-TEX-I-1D447"></use></g></g><g data-mml-node="mi" transform="translate(5233.2,0)"><use data-c="1D453" xlink:href="#MJX-83-TEX-I-1D453"></use></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi mathvariant="normal">Φ</mi><mi>T</mi></msup><mi mathvariant="normal">Φ</mi><mrow data-mjx-texclass="ORD"><mover><mi>x</mi><mo stretchy="false">^</mo></mover></mrow><mo>=</mo><msup><mi mathvariant="normal">Φ</mi><mi>T</mi></msup><mi>f</mi></math></mjx-assistive-mml></mjx-container><script type="math/tex">\Phi^T\Phi\hat{x} = \Phi^Tf</script><span>. Therefore, we can calculate </span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="1.294ex" height="1.857ex" role="img" focusable="false" viewBox="0 -810 572 821" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.025ex;"><defs><path id="MJX-84-TEX-I-1D465" d="M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z"></path><path id="MJX-84-TEX-N-5E" d="M112 560L249 694L257 686Q387 562 387 560L361 531Q359 532 303 581L250 627L195 580Q182 569 169 557T148 538L140 532Q138 530 125 546L112 560Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="TeXAtom" data-mjx-texclass="ORD"><g data-mml-node="mover"><g data-mml-node="mi"><use data-c="1D465" xlink:href="#MJX-84-TEX-I-1D465"></use></g><g data-mml-node="mo" transform="translate(313.8,16) translate(-250 0)"><use data-c="5E" xlink:href="#MJX-84-TEX-N-5E"></use></g></g></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow data-mjx-texclass="ORD"><mover><mi>x</mi><mo stretchy="false">^</mo></mover></mrow></math></mjx-assistive-mml></mjx-container><script type="math/tex">\hat{x}</script><span> as </span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg 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stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><use data-c="1D452" xlink:href="#MJX-91-TEX-I-1D452"></use></g><g data-mml-node="mi" transform="translate(466,0)"><use data-c="1D45F" xlink:href="#MJX-91-TEX-I-1D45F"></use></g><g data-mml-node="mi" transform="translate(917,0)"><use data-c="1D45F" xlink:href="#MJX-91-TEX-I-1D45F"></use></g><g data-mml-node="mi" transform="translate(1368,0)"><use data-c="1D45C" xlink:href="#MJX-91-TEX-I-1D45C"></use></g><g data-mml-node="mi" transform="translate(1853,0)"><use data-c="1D45F" xlink:href="#MJX-91-TEX-I-1D45F"></use></g><g data-mml-node="mo" transform="translate(2581.8,0)"><use data-c="3D" xlink:href="#MJX-91-TEX-N-3D"></use></g><g data-mml-node="mn" transform="translate(3637.6,0)"><use data-c="31" xlink:href="#MJX-91-TEX-N-31"></use><use data-c="33" xlink:href="#MJX-91-TEX-N-33" transform="translate(500,0)"></use><use data-c="36" xlink:href="#MJX-91-TEX-N-36" transform="translate(1000,0)"></use><use data-c="2E" xlink:href="#MJX-91-TEX-N-2E" transform="translate(1500,0)"></use><use data-c="39" xlink:href="#MJX-91-TEX-N-39" transform="translate(1778,0)"></use><use data-c="37" xlink:href="#MJX-91-TEX-N-37" transform="translate(2278,0)"></use><use data-c="32" xlink:href="#MJX-91-TEX-N-32" transform="translate(2778,0)"></use><use data-c="32" xlink:href="#MJX-91-TEX-N-32" transform="translate(3278,0)"></use></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>e</mi><mi>r</mi><mi>r</mi><mi>o</mi><mi>r</mi><mo>=</mo><mn>136.9722</mn></math></mjx-assistive-mml></mjx-container><script type="math/tex">error = 136.9722</script></p></li><li><p><span>In my opinion, the errors in task 6 and 7 cannot be compared directly due to the difference in matrix dimensions. However, I have normalized these two errors. The normalized error in task 6 is 3.125, while the normalized error in task 7 is 11.41. Therefore, the approximation in task 7 is worse compared to task 6.</span></p><p><span>  </span></p></li></ul><h2 id='8-image-bases'><span>8. Image bases</span></h2><ul><li><p><span>MATLAB code:</span></p></li></ul><pre class="md-fences md-end-block md-fences-with-lineno ty-contain-cm modeLoaded" spellcheck="false" lang="matlab" style="break-inside: unset;"><div class="CodeMirror cm-s-inner cm-s-null-scroll CodeMirror-wrap" lang="matlab"><div style="overflow: hidden; position: relative; width: 3px; height: 0px; top: 9.51875px; left: 44px;"><textarea autocorrect="off" autocapitalize="off" spellcheck="false" tabindex="0" style="position: absolute; bottom: -1em; padding: 0px; width: 1000px; height: 1em; outline: none;"></textarea></div><div class="CodeMirror-scrollbar-filler" cm-not-content="true"></div><div class="CodeMirror-gutter-filler" cm-not-content="true"></div><div class="CodeMirror-scroll" tabindex="-1"><div class="CodeMirror-sizer" style="margin-left: 36px; margin-bottom: 0px; border-right-width: 0px; padding-right: 0px; padding-bottom: 0px;"><div style="position: relative; top: 0px;"><div class="CodeMirror-lines" role="presentation"><div role="presentation" style="position: relative; outline: none;"><div class="CodeMirror-measure"><div class="CodeMirror-linenumber CodeMirror-gutter-elt"><div>56</div></div></div><div class="CodeMirror-measure"></div><div style="position: relative; z-index: 1;"></div><div class="CodeMirror-code" role="presentation" style=""><div class="CodeMirror-activeline" style="position: relative;"><div class="CodeMirror-activeline-background CodeMirror-linebackground"></div><div class="CodeMirror-gutter-background CodeMirror-activeline-gutter" style="left: -36px; width: 36px;"></div><div class="CodeMirror-gutter-wrapper CodeMirror-activeline-gutter" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt CodeMirror-linenumber-show" style="left: 0px; width: 27px;">1</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-comment">%% main</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">2</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-builtin">clear</span>;</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">3</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-variable">clc</span>;</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">4</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-variable">close</span> <span class="cm-variable">all</span>;</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">5</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span cm-text="" cm-zwsp="">
</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">6</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-variable">DISPLAY</span> = <span class="cm-number">0</span>;</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">7</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-variable">load</span>(<span class="cm-string">'assignment1bases.mat'</span>)</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">8</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span cm-text="" cm-zwsp="">
</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">9</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-comment">%%%</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt CodeMirror-linenumber-show" style="left: 0px; width: 27px;">10</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-comment">% Due to space limitations, the initialization part is not included here.</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">11</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-comment">%%%</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">12</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span cm-text="" cm-zwsp="">
</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">13</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-keyword">for</span> <span class="cm-variable">j</span>=<span class="cm-number">1</span>:<span class="cm-number">2</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">14</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp;<span class="cm-keyword">for</span> <span class="cm-variable">i</span>=<span class="cm-number">1</span>:<span class="cm-number">400</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">15</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp; &nbsp; &nbsp;<span class="cm-variable">img</span> = <span class="cm-variable">stacks</span>{<span class="cm-variable">j</span>}(:,:,<span class="cm-variable">i</span>);</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">16</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" class="cm-tab-wrap-hack" style="padding-right: 0.1px;"><span class="cm-tab" role="presentation" cm-text="	">    </span><span class="cm-tab" role="presentation" cm-text="	">    </span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">17</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-tab" role="presentation" cm-text="	">    </span><span class="cm-tab" role="presentation" cm-text="	">    </span><span class="cm-comment">% Finding the approximate image and error.</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">18</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp; &nbsp;  [<span class="cm-variable">img_hat1</span>, <span class="cm-variable">error1</span>] = <span class="cm-variable">PCA</span>(<span class="cm-variable">img</span>, <span class="cm-variable">bases1</span>);</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">19</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp; &nbsp;  [<span class="cm-variable">img_hat2</span>, <span class="cm-variable">error2</span>] = <span class="cm-variable">PCA</span>(<span class="cm-variable">img</span>, <span class="cm-variable">bases2</span>);</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt CodeMirror-linenumber-show" style="left: 0px; width: 27px;">20</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp; &nbsp;  [<span class="cm-variable">img_hat3</span>, <span class="cm-variable">error3</span>] = <span class="cm-variable">PCA</span>(<span class="cm-variable">img</span>, <span class="cm-variable">bases3</span>);</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">21</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" class="cm-tab-wrap-hack" style="padding-right: 0.1px;"><span class="cm-tab" role="presentation" cm-text="	">    </span><span class="cm-tab" role="presentation" cm-text="	">    </span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">22</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-tab" role="presentation" cm-text="	">    </span><span class="cm-tab" role="presentation" cm-text="	">    </span><span class="cm-comment">% sum the errors</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">23</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp; &nbsp; &nbsp;<span class="cm-variable">e1_sum</span> = <span class="cm-variable">e1_sum</span> <span class="cm-operator">+</span> <span class="cm-variable">error1</span>;</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">24</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp; &nbsp; &nbsp;<span class="cm-variable">e2_sum</span> = <span class="cm-variable">e2_sum</span> <span class="cm-operator">+</span> <span class="cm-variable">error2</span>;</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">25</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp; &nbsp; &nbsp;<span class="cm-variable">e3_sum</span> = <span class="cm-variable">e3_sum</span> <span class="cm-operator">+</span> <span class="cm-variable">error3</span>;</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">26</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp; &nbsp; &nbsp;</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">27</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp; &nbsp; &nbsp;<span class="cm-comment">% Display images</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">28</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp; &nbsp; &nbsp;<span class="cm-keyword">if</span> <span class="cm-variable">DISPLAY</span> <span class="cm-operator">==</span> <span class="cm-number">1</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">29</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;<span class="cm-variable">figure</span>;</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt CodeMirror-linenumber-show" style="left: 0px; width: 27px;">30</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;<span class="cm-variable">subplot</span>(<span class="cm-number">4</span>, <span class="cm-number">1</span>, <span class="cm-number">1</span>);</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">31</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;<span class="cm-variable">imshow</span>(<span class="cm-variable">uint8</span>(<span class="cm-variable">img</span>));</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">32</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;<span class="cm-builtin">title</span>(<span class="cm-string">'Image'</span>);</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">33</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp;</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">34</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;<span class="cm-variable">subplot</span>(<span class="cm-number">4</span>, <span class="cm-number">1</span>, <span class="cm-number">2</span>);</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">35</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;<span class="cm-variable">imshow</span>(<span class="cm-variable">uint8</span>(<span class="cm-variable">img_hat1</span>));</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">36</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;<span class="cm-builtin">title</span>(<span class="cm-string">'Image Base 1'</span>);</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">37</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp;</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">38</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;<span class="cm-variable">subplot</span>(<span class="cm-number">4</span>, <span class="cm-number">1</span>, <span class="cm-number">3</span>);</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">39</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;<span class="cm-variable">imshow</span>(<span class="cm-variable">uint8</span>(<span class="cm-variable">img_hat2</span>));</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt CodeMirror-linenumber-show" style="left: 0px; width: 27px;">40</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;<span class="cm-builtin">title</span>(<span class="cm-string">'Image Base 2'</span>);</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">41</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp;</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">42</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;<span class="cm-variable">subplot</span>(<span class="cm-number">4</span>, <span class="cm-number">1</span>, <span class="cm-number">4</span>);</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">43</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;<span class="cm-variable">imshow</span>(<span class="cm-variable">uint8</span>(<span class="cm-variable">img_hat3</span>));</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">44</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;<span class="cm-builtin">title</span>(<span class="cm-string">'Image Base 3'</span>);</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">45</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp; &nbsp; &nbsp;<span class="cm-keyword">end</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">46</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span cm-text="" cm-zwsp="">
</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">47</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp;<span class="cm-keyword">end</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">48</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" class="cm-tab-wrap-hack" style="padding-right: 0.1px;"><span class="cm-tab" role="presentation" cm-text="	">    </span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">49</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-tab" role="presentation" cm-text="	">    </span><span class="cm-comment">% caculate the mean errors</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt CodeMirror-linenumber-show" style="left: 0px; width: 27px;">50</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp;<span class="cm-builtin">mean</span>(<span class="cm-variable">j</span>,<span class="cm-number">1</span>) = <span class="cm-variable">e1_sum</span><span class="cm-operator">/</span><span class="cm-number">400</span>;</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">51</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp;<span class="cm-builtin">mean</span>(<span class="cm-variable">j</span>,<span class="cm-number">2</span>) = <span class="cm-variable">e2_sum</span><span class="cm-operator">/</span><span class="cm-number">400</span>;</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">52</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp;<span class="cm-builtin">mean</span>(<span class="cm-variable">j</span>,<span class="cm-number">3</span>) = <span class="cm-variable">e3_sum</span><span class="cm-operator">/</span><span class="cm-number">400</span>;</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">53</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span cm-text="" cm-zwsp="">
</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">54</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-keyword">end</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">55</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span cm-text="" cm-zwsp="">
</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt CodeMirror-linenumber-show" style="left: 0px; width: 27px;">56</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-builtin">mean</span></span></pre></div></div></div></div></div></div><div style="position: absolute; height: 0px; width: 1px; border-bottom: 0px solid transparent; top: 1290px;"></div><div class="CodeMirror-gutters" style="height: 1290px; left: 0px;"><div class="CodeMirror-gutter CodeMirror-linenumbers" style="width: 35px;"></div></div></div></div></pre><pre class="md-fences md-end-block md-fences-with-lineno ty-contain-cm modeLoaded" spellcheck="false" lang="matlab"><div class="CodeMirror cm-s-inner cm-s-null-scroll CodeMirror-wrap" lang="matlab"><div style="overflow: hidden; position: relative; width: 3px; height: 0px; top: 9.51875px; left: 44px;"><textarea autocorrect="off" autocapitalize="off" spellcheck="false" tabindex="0" style="position: absolute; bottom: -1em; padding: 0px; width: 1000px; height: 1em; outline: none;"></textarea></div><div class="CodeMirror-scrollbar-filler" cm-not-content="true"></div><div class="CodeMirror-gutter-filler" cm-not-content="true"></div><div class="CodeMirror-scroll" tabindex="-1"><div class="CodeMirror-sizer" style="margin-left: 36px; margin-bottom: 0px; border-right-width: 0px; padding-right: 0px; padding-bottom: 0px;"><div style="position: relative; top: 0px;"><div class="CodeMirror-lines" role="presentation"><div role="presentation" style="position: relative; outline: none;"><div class="CodeMirror-measure"><div class="CodeMirror-linenumber CodeMirror-gutter-elt"><div>13</div></div></div><div class="CodeMirror-measure"></div><div style="position: relative; z-index: 1;"></div><div class="CodeMirror-code" role="presentation" style=""><div class="CodeMirror-activeline" style="position: relative;"><div class="CodeMirror-activeline-background CodeMirror-linebackground"></div><div class="CodeMirror-gutter-background CodeMirror-activeline-gutter" style="left: -36px; width: 36px;"></div><div class="CodeMirror-gutter-wrapper CodeMirror-activeline-gutter" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt CodeMirror-linenumber-show" style="left: 0px; width: 27px;">1</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-builtin">function</span> [<span class="cm-variable">img_hat</span>, <span class="cm-builtin">error</span>] = <span class="cm-variable">PCA</span>(<span class="cm-variable">img</span>, <span class="cm-variable">bases</span>)</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">2</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span cm-text="" cm-zwsp="">
</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">3</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-tab" role="presentation" cm-text="	">    </span><span class="cm-comment">% Solve for linear combination coefficients</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">4</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp;<span class="cm-variable">coefficients</span> = <span class="cm-variable">bases</span> <span class="cm-operator">\</span> <span class="cm-variable">img</span>(:);</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">5</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span cm-text="" cm-zwsp="">
</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">6</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp;<span class="cm-comment">% Reconstruct vector</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">7</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp;<span class="cm-variable">img_hat</span> = <span class="cm-variable">bases</span> <span class="cm-operator">*</span> <span class="cm-variable">coefficients</span>;</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">8</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span cm-text="" cm-zwsp="">
</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">9</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp;<span class="cm-comment">% Restore the reconstructed vector to its original shape</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt CodeMirror-linenumber-show" style="left: 0px; width: 27px;">10</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp;<span class="cm-variable">img_hat</span> = <span class="cm-builtin">reshape</span>(<span class="cm-variable">img_hat</span>, <span class="cm-builtin">size</span>(<span class="cm-variable">img</span>));</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">11</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span cm-text="" cm-zwsp="">
</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">12</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp;<span class="cm-builtin">error</span> = <span class="cm-builtin">norm</span>(<span class="cm-variable">img_hat</span><span class="cm-operator">-</span><span class="cm-variable">img</span>, <span class="cm-number">2</span>);</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt CodeMirror-linenumber-show" style="left: 0px; width: 27px;">13</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-keyword">end</span></span></pre></div></div></div></div></div></div><div style="position: absolute; height: 0px; width: 1px; border-bottom: 0px solid transparent; top: 299px;"></div><div class="CodeMirror-gutters" style="height: 299px; left: 0px;"><div class="CodeMirror-gutter CodeMirror-linenumbers" style="width: 35px;"></div></div></div></div></pre><ul><li><p><span>Obviously, the images in the first test set are part of people&#39;s faces and the second test set are pictures of people&#39;s faces. The first set of basis element images represents the variations in lightness and darkness in the original image. The third set is similar to the first one, but it only has four regions. The basis element images appear to be of higher quality compared to the others as they show slightly blurred faces, but they are mostly similar to each other. Unlike the original image, the main similarity lies in the variations of lightness and darkness.</span></p><div align=center><img src = "E:\隆德大学LUND\2023 period 1\Image Analysis\Assignment 1\figure\WPS拼图1.png" style="zoom:60%;"></div></li></ul><div align=center><img src = "E:\隆德大学LUND\2023 period 1\Image Analysis\Assignment 1\figure\WPS拼图0.png" style="zoom:60%;"></div><ul><li><p><span>Mean of the error norms:</span></p><p><img src = "C:\Users\30379\AppData\Roaming\Typora\typora-user-images\image-20230908150126825.png" style="zoom:60%;"></p></li><li><p><span>In my opinion, the first basis is the best for test set 1. It has the smallest mean error among the three bases. Additionally, the second basis does not accurately fits the local features of people&#39;s faces, as it mainly represents the overall face.</span></p></li><li><p><span>The second basis is the best for test set 2. Although it does not have the smallest mean error, it fits the overall face better.</span></p></li></ul><p>&nbsp;</p></div></div>
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